# lfunc_search downloaded from the LMFDB on 27 June 2026. # Search link: https://www.lmfdb.org/L/rational/2/1920/120.29 # Query "{'degree': 2, 'conductor': 1920, 'rational': True}" returned 44 lfunc_searchs, sorted by root analytic conductor. # Each entry in the following data list has the form: # [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, $\epsilon$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "2-1920-120.29-c0-0-1" 0.9788793716568782 0.9582048242553647 2 1920 "120.29" [] [[0.0, 0.0]] 0 true 1 0 0.853544766397543477419339193187 ["ModularForm/GL2/Q/holomorphic/1920/1/i/a/449/1", "ModularForm/GL2/Q/holomorphic/1920/1/i/a", "ArtinRepresentation/2.1920.4t3.l.a", "ArtinRepresentation/2.1920.4t3.l"] "2-1920-120.29-c0-0-3" 0.9788793716568782 0.9582048242553647 2 1920 "120.29" [] [[0.0, 0.0]] 0 true 1 0 1.16006904794089283083604072702 ["ModularForm/GL2/Q/holomorphic/1920/1/i/b/449/1", "ModularForm/GL2/Q/holomorphic/1920/1/i/b", "ArtinRepresentation/2.1920.4t3.i.a", "ArtinRepresentation/2.1920.4t3.i"] "2-1920-120.29-c0-0-4" 0.9788793716568782 0.9582048242553647 2 1920 "120.29" [] [[0.0, 0.0]] 0 true 1 0 1.28822101635495953522537599373 ["ModularForm/GL2/Q/holomorphic/1920/1/i/c/449/1", "ModularForm/GL2/Q/holomorphic/1920/1/i/c", "ArtinRepresentation/2.1920.4t3.k.a", "ArtinRepresentation/2.1920.4t3.k"] "2-1920-120.29-c0-0-6" 0.9788793716568782 0.9582048242553647 2 1920 "120.29" [] [[0.0, 0.0]] 0 true 1 0 1.67444121606388394473781174539 ["ModularForm/GL2/Q/holomorphic/1920/1/i/d/449/1", "ModularForm/GL2/Q/holomorphic/1920/1/i/d", "ArtinRepresentation/2.1920.4t3.j.a", "ArtinRepresentation/2.1920.4t3.j"] "2-1920-1.1-c1-0-0" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.55094847574877251324997839517 ["EllipticCurve/Q/1920/b", "ModularForm/GL2/Q/holomorphic/1920/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/b"] "2-1920-1.1-c1-0-1" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.60908499094619446828089812922 ["EllipticCurve/Q/1920/a", "ModularForm/GL2/Q/holomorphic/1920/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/a"] "2-1920-1.1-c1-0-10" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.06318891316584379720882015896 ["EllipticCurve/Q/1920/f", "ModularForm/GL2/Q/holomorphic/1920/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/f"] "2-1920-1.1-c1-0-11" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.09505201742716397244922543371 ["EllipticCurve/Q/1920/r", "ModularForm/GL2/Q/holomorphic/1920/2/a/r/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/r"] "2-1920-1.1-c1-0-12" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.09595735816982629557946257255 ["EllipticCurve/Q/1920/l", "ModularForm/GL2/Q/holomorphic/1920/2/a/l/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/l"] "2-1920-1.1-c1-0-13" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.10173831482498274651074899895 ["EllipticCurve/Q/1920/v", "ModularForm/GL2/Q/holomorphic/1920/2/a/v/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/v"] "2-1920-1.1-c1-0-14" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.10471661201841061507928486136 ["EllipticCurve/Q/1920/q", "ModularForm/GL2/Q/holomorphic/1920/2/a/q/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/q"] "2-1920-1.1-c1-0-15" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.12397443494594220839491514104 ["EllipticCurve/Q/1920/u", "ModularForm/GL2/Q/holomorphic/1920/2/a/u/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/u"] "2-1920-1.1-c1-0-16" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.15729587229321409219878111918 ["EllipticCurve/Q/1920/x", "ModularForm/GL2/Q/holomorphic/1920/2/a/x/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/x"] "2-1920-1.1-c1-0-18" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.22056588467062317603572071417 ["EllipticCurve/Q/1920/w", "ModularForm/GL2/Q/holomorphic/1920/2/a/w/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/w"] "2-1920-1.1-c1-0-20" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.44640173159626840226202689232 ["EllipticCurve/Q/1920/c", "ModularForm/GL2/Q/holomorphic/1920/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/c"] "2-1920-1.1-c1-0-22" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.52561957778558510406501144135 ["EllipticCurve/Q/1920/g", "ModularForm/GL2/Q/holomorphic/1920/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/g"] "2-1920-1.1-c1-0-23" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.52877886661374831253428270259 ["EllipticCurve/Q/1920/e", "ModularForm/GL2/Q/holomorphic/1920/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/e"] "2-1920-1.1-c1-0-24" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.55325096018789847060637121011 ["EllipticCurve/Q/1920/h", "ModularForm/GL2/Q/holomorphic/1920/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/h"] "2-1920-1.1-c1-0-26" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.68720638789308651755146426845 ["EllipticCurve/Q/1920/j", "ModularForm/GL2/Q/holomorphic/1920/2/a/j/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/j"] "2-1920-1.1-c1-0-27" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.73508381507895261886045237038 ["EllipticCurve/Q/1920/k", "ModularForm/GL2/Q/holomorphic/1920/2/a/k/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/k"] "2-1920-1.1-c1-0-28" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.76828397665908917268536961976 ["EllipticCurve/Q/1920/n", "ModularForm/GL2/Q/holomorphic/1920/2/a/n/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/n"] "2-1920-1.1-c1-0-29" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.81084238627342889349146058738 ["EllipticCurve/Q/1920/o", "ModularForm/GL2/Q/holomorphic/1920/2/a/o/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/o"] "2-1920-1.1-c1-0-3" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.74394243212741429594461253345 ["EllipticCurve/Q/1920/d", "ModularForm/GL2/Q/holomorphic/1920/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/d"] "2-1920-1.1-c1-0-30" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.84416625898130938268214518612 ["EllipticCurve/Q/1920/s", "ModularForm/GL2/Q/holomorphic/1920/2/a/s/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/s"] "2-1920-1.1-c1-0-31" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true -1 1 1.91529347433011582491924549952 ["EllipticCurve/Q/1920/t", "ModularForm/GL2/Q/holomorphic/1920/2/a/t/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/t"] "2-1920-1.1-c1-0-5" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.803706848118763804248425590938 ["EllipticCurve/Q/1920/m", "ModularForm/GL2/Q/holomorphic/1920/2/a/m/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/m"] "2-1920-1.1-c1-0-6" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 0.951853117585691762831532111621 ["EllipticCurve/Q/1920/i", "ModularForm/GL2/Q/holomorphic/1920/2/a/i/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/i"] "2-1920-1.1-c1-0-9" 3.9155174866275138 15.331277188085842 2 1920 "1.1" [] [[0.5, 0.0]] 1 true 1 0 1.05573399756661793291429489696 ["EllipticCurve/Q/1920/p", "ModularForm/GL2/Q/holomorphic/1920/2/a/p/1/1", "ModularForm/GL2/Q/holomorphic/1920/2/a/p"] "2-1920-1.1-c3-0-12" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true 1 0 0.55685902670791633208141337830 ["ModularForm/GL2/Q/holomorphic/1920/4/a/h/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/h"] "2-1920-1.1-c3-0-26" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true 1 0 0.71092078837305309168404239231 ["ModularForm/GL2/Q/holomorphic/1920/4/a/e/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/e"] "2-1920-1.1-c3-0-34" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true 1 0 0.792174344149284531120766117444 ["ModularForm/GL2/Q/holomorphic/1920/4/a/c/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/c"] "2-1920-1.1-c3-0-56" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.08224957398198554207873136015 ["ModularForm/GL2/Q/holomorphic/1920/4/a/a/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/a"] "2-1920-1.1-c3-0-62" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.16787284993251330959563082653 ["ModularForm/GL2/Q/holomorphic/1920/4/a/d/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/d"] "2-1920-1.1-c3-0-65" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.21253030918790524782857233264 ["ModularForm/GL2/Q/holomorphic/1920/4/a/g/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/g"] "2-1920-1.1-c3-0-68" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.23040477083167522771324681737 ["ModularForm/GL2/Q/holomorphic/1920/4/a/f/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/f"] "2-1920-1.1-c3-0-7" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true 1 0 0.44846442092585047321767292918 ["ModularForm/GL2/Q/holomorphic/1920/4/a/b/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/b"] "2-1920-1.1-c3-0-70" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.25278429442528764009786742038 ["ModularForm/GL2/Q/holomorphic/1920/4/a/i/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/i"] "2-1920-1.1-c3-0-74" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.27881225111629509680979135144 ["ModularForm/GL2/Q/holomorphic/1920/4/a/j/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/j"] "2-1920-1.1-c3-0-82" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.38332612917836187612635227665 ["ModularForm/GL2/Q/holomorphic/1920/4/a/k/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/k"] "2-1920-1.1-c3-0-83" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.40091551145768825362113310234 ["ModularForm/GL2/Q/holomorphic/1920/4/a/l/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/l"] "2-1920-1.1-c3-0-87" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.45093381998494233024173577530 ["ModularForm/GL2/Q/holomorphic/1920/4/a/m/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/m"] "2-1920-1.1-c3-0-90" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.47150603134322911030503751168 ["ModularForm/GL2/Q/holomorphic/1920/4/a/o/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/o"] "2-1920-1.1-c3-0-93" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.54588126413334418039147974213 ["ModularForm/GL2/Q/holomorphic/1920/4/a/n/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/n"] "2-1920-1.1-c3-0-94" 10.643480032913203 113.28366721102203 2 1920 "1.1" [] [[1.5, 0.0]] 3 true -1 1 1.60150371250202377886872948956 ["ModularForm/GL2/Q/holomorphic/1920/4/a/p/1/1", "ModularForm/GL2/Q/holomorphic/1920/4/a/p"] # Label -- # Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where # * $d$ is the degree of $L$. # * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions. # * q.k is the label of the primitive Dirichlet character from which the central character is induced. # * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation. # * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label. #$\alpha$ (root_analytic_conductor) -- # If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields. #$A$ (analytic_conductor) -- # The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number # \[ # A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N. # \] #$d$ (degree) -- # The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor. #$N$ (conductor) -- # The **conductor** of an L-function is the integer $N$ occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function # associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$. # In the literature, the word _level_ is sometimes used instead of _conductor_. #$\chi$ (central_character) -- # An L-function has an Euler product of the form # $L(s) = \prod_p L_p(p^{-s})^{-1}$ # where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function. # Here, $N$ is the conductor of $L$. #$\mu$ (mus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$\nu$ (nus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$w$ (motivic_weight) -- # The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$. # If the L-function arises from a motive, then the weight of the motive has the # same parity as the motivic weight of the L-function, but the weight of the motive # could be larger. This apparent discrepancy comes from the fact that a Tate twist # increases the weight of the motive. This corresponds to the change of variables # $s \mapsto s + j$ in the L-function of the motive. #prim (primitive) -- # An L-function is primitive if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$. #$\epsilon$ (root_number) -- # The **sign** of the functional equation of an analytic L-function, also called the **root number**, is the complex number $\varepsilon$ that appears in the functional equation of $\Lambda(s)=\varepsilon \overline{\Lambda}(1-s)$. The sign appears as the 4th entry in the quadruple # known as the Selberg data. #$r$ (order_of_vanishing) -- # The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point. # When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point). #First zero (z1) -- # The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$. # Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization). # The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number. #Origin (instance_urls) -- # L-functions arise from many different sources. Already in degree 2 we have examples of # L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions). # Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).