# lfunc_search downloaded from the LMFDB on 12 April 2026. # Search link: https://www.lmfdb.org/L/2/42/7.5/c2-0 # Query "{'degree': 2, 'conductor': 42, 'spectral_label': 'c2-0'}" returned 24 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, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "2-42-21.11-c2-0-0" 1.0697743268153557 1.1444171103132472 2 42 "21.11" [] [[1.0, 0.0]] 2 true true false false -0.4050319016762627 0 0.67547929323321649937980912680 ["ModularForm/GL2/Q/holomorphic/42/3/h/b/11/1"] "2-42-21.11-c2-0-1" 1.0697743268153557 1.1444171103132472 2 42 "21.11" [] [[1.0, 0.0]] 2 true true false false -0.04554853592115429 0 2.36855596773987885080361239393 ["ModularForm/GL2/Q/holomorphic/42/3/h/b/11/2"] "2-42-21.11-c2-0-2" 1.0697743268153557 1.1444171103132472 2 42 "21.11" [] [[1.0, 0.0]] 2 true true false false -0.0077304283528976045 0 2.66180964612968656785158117705 ["ModularForm/GL2/Q/holomorphic/42/3/h/b/11/4"] "2-42-21.11-c2-0-3" 1.0697743268153557 1.1444171103132472 2 42 "21.11" [] [[1.0, 0.0]] 2 true true false false 0.14750653384500276 0 2.80747555164711503271266000506 ["ModularForm/GL2/Q/holomorphic/42/3/h/a/11/1"] "2-42-21.11-c2-0-4" 1.0697743268153557 1.1444171103132472 2 42 "21.11" [] [[1.0, 0.0]] 2 true true false false 0.03933308587561003 0 3.24154603880256607681780776282 ["ModularForm/GL2/Q/holomorphic/42/3/h/a/11/2"] "2-42-21.11-c2-0-5" 1.0697743268153557 1.1444171103132472 2 42 "21.11" [] [[1.0, 0.0]] 2 true true false false 0.16532343872487346 0 3.83322646708001216848073302714 ["ModularForm/GL2/Q/holomorphic/42/3/h/b/11/3"] "2-42-21.2-c2-0-0" 1.0697743268153557 1.1444171103132472 2 42 "21.2" [] [[1.0, 0.0]] 2 true true false false -0.14750653384500276 0 1.39112087438542918662457635971 ["ModularForm/GL2/Q/holomorphic/42/3/h/a/23/1"] "2-42-21.2-c2-0-1" 1.0697743268153557 1.1444171103132472 2 42 "21.2" [] [[1.0, 0.0]] 2 true true false false -0.16532343872487346 0 2.12843528403547952178587957365 ["ModularForm/GL2/Q/holomorphic/42/3/h/b/23/3"] "2-42-21.2-c2-0-2" 1.0697743268153557 1.1444171103132472 2 42 "21.2" [] [[1.0, 0.0]] 2 true true false false 0.04554853592115429 0 2.47624137765638921638372235460 ["ModularForm/GL2/Q/holomorphic/42/3/h/b/23/2"] "2-42-21.2-c2-0-3" 1.0697743268153557 1.1444171103132472 2 42 "21.2" [] [[1.0, 0.0]] 2 true true false false 0.0077304283528976045 0 3.01201525965927102144182932257 ["ModularForm/GL2/Q/holomorphic/42/3/h/b/23/4"] "2-42-21.2-c2-0-4" 1.0697743268153557 1.1444171103132472 2 42 "21.2" [] [[1.0, 0.0]] 2 true true false false -0.03933308587561003 0 3.21653654134617585231728888020 ["ModularForm/GL2/Q/holomorphic/42/3/h/a/23/2"] "2-42-21.2-c2-0-5" 1.0697743268153557 1.1444171103132472 2 42 "21.2" [] [[1.0, 0.0]] 2 true true false false 0.4050319016762627 0 3.91588356256673611795640862293 ["ModularForm/GL2/Q/holomorphic/42/3/h/b/23/1"] "2-42-3.2-c2-0-0" 1.0697743268153557 1.1444171103132472 2 42 "3.2" [] [[1.0, 0.0]] 2 true true false false -0.32812640472793253 0 1.23197583297330044693123404601 ["ModularForm/GL2/Q/holomorphic/42/3/b/a/29/3"] "2-42-3.2-c2-0-1" 1.0697743268153557 1.1444171103132472 2 42 "3.2" [] [[1.0, 0.0]] 2 true true false false -0.1718735952720675 0 2.25442640976318514445653685193 ["ModularForm/GL2/Q/holomorphic/42/3/b/a/29/4"] "2-42-3.2-c2-0-2" 1.0697743268153557 1.1444171103132472 2 42 "3.2" [] [[1.0, 0.0]] 2 true true false false 0.1718735952720675 0 3.44301175405416403260060461427 ["ModularForm/GL2/Q/holomorphic/42/3/b/a/29/2"] "2-42-3.2-c2-0-3" 1.0697743268153557 1.1444171103132472 2 42 "3.2" [] [[1.0, 0.0]] 2 true true false false 0.32812640472793253 0 4.00088304407135574630311318648 ["ModularForm/GL2/Q/holomorphic/42/3/b/a/29/1"] "2-42-7.3-c2-0-0" 1.0697743268153557 1.1444171103132472 2 42 "7.3" [] [[1.0, 0.0]] 2 true true false false -0.14679709583952122 0 2.36411800528140250691883631837 ["ModularForm/GL2/Q/holomorphic/42/3/g/a/31/2"] "2-42-7.3-c2-0-1" 1.0697743268153557 1.1444171103132472 2 42 "7.3" [] [[1.0, 0.0]] 2 true true false false 0.10645118973162895 0 2.72031556201279849766172301524 ["ModularForm/GL2/Q/holomorphic/42/3/g/a/31/1"] "2-42-7.5-c2-0-0" 1.0697743268153557 1.1444171103132472 2 42 "7.5" [] [[1.0, 0.0]] 2 true true false false -0.10645118973162895 0 2.20604359483068962765831611063 ["ModularForm/GL2/Q/holomorphic/42/3/g/a/19/1"] "2-42-7.5-c2-0-1" 1.0697743268153557 1.1444171103132472 2 42 "7.5" [] [[1.0, 0.0]] 2 true true false false 0.14679709583952122 0 3.52432439487018561702569581882 ["ModularForm/GL2/Q/holomorphic/42/3/g/a/19/2"] "2-42-7.6-c2-0-0" 1.0697743268153557 1.1444171103132472 2 42 "7.6" [] [[1.0, 0.0]] 2 true true false false -0.175280691521465 0 1.47387573767770102315147410442 ["ModularForm/GL2/Q/holomorphic/42/3/c/a/13/2"] "2-42-7.6-c2-0-1" 1.0697743268153557 1.1444171103132472 2 42 "7.6" [] [[1.0, 0.0]] 2 true true false false -0.05190483430704006 0 2.65424106596318642352946289616 ["ModularForm/GL2/Q/holomorphic/42/3/c/a/13/4"] "2-42-7.6-c2-0-2" 1.0697743268153557 1.1444171103132472 2 42 "7.6" [] [[1.0, 0.0]] 2 true true false false 0.175280691521465 0 2.93119924689143621118693881153 ["ModularForm/GL2/Q/holomorphic/42/3/c/a/13/1"] "2-42-7.6-c2-0-3" 1.0697743268153557 1.1444171103132472 2 42 "7.6" [] [[1.0, 0.0]] 2 true true false false 0.05190483430704006 0 3.31428425515721586056396944495 ["ModularForm/GL2/Q/holomorphic/42/3/c/a/13/3"] # 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 $1$. #arith (algebraic) -- # An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers. #$\mathbb{Q}$ (rational) -- # A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes # \[ # L(s)=\prod_pL_p(p^{-s})^{-1} # \] # with $L_p\in \Z[T]$. #self-dual (self_dual) -- # An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real. #$\operatorname{Arg}(\epsilon)$ (root_angle) -- # The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$. #$r$ (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).