# lfunc_search downloaded from the LMFDB on 25 May 2026. # Search link: https://www.lmfdb.org/L/2/1805/95.69/c0-0 # Query "{'degree': 2, 'conductor': 1805, 'spectral_label': 'c0-0'}" returned 42 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. "2-1805-5.2-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "5.2" [] [[0.0, 0.0]] 0 true true false false -0.0881040955873917 0 0.860885543085242964320735710058 ["ModularForm/GL2/Q/holomorphic/1805/1/f/a/362/1"] "2-1805-5.3-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "5.3" [] [[0.0, 0.0]] 0 true true false false 0.0881040955873917 0 1.47383051983270950004399722448 ["ModularForm/GL2/Q/holomorphic/1805/1/f/a/723/1"] "2-1805-95.14-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.14" [] [[0.0, 0.0]] 0 true true false false 0.41046391945430255 0 0.30988814033562018342300677053 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/299/1"] "2-1805-95.14-c0-0-1" 0.9491113463669947 0.9008123478025694 2 1805 "95.14" [] [[0.0, 0.0]] 0 true true false false -0.08953608054569746 0 1.27942351034480932263572465497 ["ModularForm/GL2/Q/holomorphic/1805/1/o/a/299/1"] "2-1805-95.14-c0-0-2" 0.9491113463669947 0.9008123478025694 2 1805 "95.14" [] [[0.0, 0.0]] 0 true true false false 0.41046391945430255 0 2.57663511492882243031011014158 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/299/2"] "2-1805-95.17-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.17" [] [[0.0, 0.0]] 0 true true false false 0.0544719973042897 0 1.49717131690372015279730099094 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/967/1"] "2-1805-95.23-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.23" [] [[0.0, 0.0]] 0 true true false false -0.44828809165331834 0 0.53367304276813458563710460650 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/1543/1"] "2-1805-95.28-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.28" [] [[0.0, 0.0]] 0 true true false false -0.0544719973042897 0 1.06209563989439886169501730479 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/28/1"] "2-1805-95.29-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.29" [] [[0.0, 0.0]] 0 true true false false 0.4419289012348729 0 0.936159941224155285752364785320 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/694/2"] "2-1805-95.29-c0-0-1" 0.9491113463669947 0.9008123478025694 2 1805 "95.29" [] [[0.0, 0.0]] 0 true true false false -0.05807109876512718 0 1.05990217004758937017361241634 ["ModularForm/GL2/Q/holomorphic/1805/1/o/a/694/1"] "2-1805-95.29-c0-0-2" 0.9491113463669947 0.9008123478025694 2 1805 "95.29" [] [[0.0, 0.0]] 0 true true false false 0.4419289012348729 0 2.08485033141097938306299473611 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/694/1"] "2-1805-95.34-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.34" [] [[0.0, 0.0]] 0 true true false false -0.41046391945430255 0 1.01337289331048794194138173222 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/984/2"] "2-1805-95.34-c0-0-1" 0.9491113463669947 0.9008123478025694 2 1805 "95.34" [] [[0.0, 0.0]] 0 true true false false 0.08953608054569746 0 1.34174102720537357181234126367 ["ModularForm/GL2/Q/holomorphic/1805/1/o/a/984/1"] "2-1805-95.34-c0-0-2" 0.9491113463669947 0.9008123478025694 2 1805 "95.34" [] [[0.0, 0.0]] 0 true true false false -0.41046391945430255 0 1.53668442124365781537246976097 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/984/1"] "2-1805-95.42-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.42" [] [[0.0, 0.0]] 0 true true false false 0.3755037171718983 0 1.91634048152943944396380247678 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/1182/1"] "2-1805-95.43-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.43" [] [[0.0, 0.0]] 0 true true false false -0.3755037171718983 0 0.34005321573832572035404083816 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/423/1"] "2-1805-95.47-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.47" [] [[0.0, 0.0]] 0 true true false false -0.23068018847907307 0 0.981078919682957247687741209926 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/1472/1"] "2-1805-95.59-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.59" [] [[0.0, 0.0]] 0 true true false false -0.4419289012348729 0 1.20985613021069030088407408016 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/1199/1"] "2-1805-95.59-c0-0-1" 0.9491113463669947 0.9008123478025694 2 1805 "95.59" [] [[0.0, 0.0]] 0 true true false false 0.05807109876512718 0 1.50995150883724228360282460664 ["ModularForm/GL2/Q/holomorphic/1805/1/o/a/1199/1"] "2-1805-95.59-c0-0-2" 0.9491113463669947 0.9008123478025694 2 1805 "95.59" [] [[0.0, 0.0]] 0 true true false false -0.4419289012348729 0 1.89668163392405352790239876121 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/1199/2"] "2-1805-95.62-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.62" [] [[0.0, 0.0]] 0 true true false false 0.44828809165331834 0 2.24014108874784690126845988077 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/62/1"] "2-1805-95.63-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.63" [] [[0.0, 0.0]] 0 true true false false 0.03265242099083963 0 1.53763349035060186651787064285 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/1678/1"] "2-1805-95.68-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.68" [] [[0.0, 0.0]] 0 true true false false 0.15585738364966847 0 1.90555224068989314331050168099 ["ModularForm/GL2/Q/holomorphic/1805/1/m/a/68/1"] "2-1805-95.69-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.69" [] [[0.0, 0.0]] 0 true true false false -0.13279002263552828 0 0.822938222305756065845047345046 ["ModularForm/GL2/Q/holomorphic/1805/1/h/b/69/1"] "2-1805-95.69-c0-0-1" 0.9491113463669947 0.9008123478025694 2 1805 "95.69" [] [[0.0, 0.0]] 0 true true false false -0.13279002263552828 0 1.31679355163291657121485635824 ["ModularForm/GL2/Q/holomorphic/1805/1/h/b/69/2"] "2-1805-95.69-c0-0-2" 0.9491113463669947 0.9008123478025694 2 1805 "95.69" [] [[0.0, 0.0]] 0 true true false false 0.36720997736447175 0 2.25804372038258706175905864861 ["ModularForm/GL2/Q/holomorphic/1805/1/h/a/69/1"] "2-1805-95.7-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.7" [] [[0.0, 0.0]] 0 true true false false -0.15585738364966847 0 1.36290705284242985829644560132 ["ModularForm/GL2/Q/holomorphic/1805/1/m/a/292/1"] "2-1805-95.73-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.73" [] [[0.0, 0.0]] 0 true true false false 0.14355577018394375 0 1.15973290705251031567623264986 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/1498/1"] "2-1805-95.79-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.79" [] [[0.0, 0.0]] 0 true true false false -0.1690783289172348 0 0.54542252911770121957812014451 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/1029/1"] "2-1805-95.79-c0-0-1" 0.9491113463669947 0.9008123478025694 2 1805 "95.79" [] [[0.0, 0.0]] 0 true true false false -0.1690783289172348 0 1.74604162140077942171069522001 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/1029/2"] "2-1805-95.79-c0-0-2" 0.9491113463669947 0.9008123478025694 2 1805 "95.79" [] [[0.0, 0.0]] 0 true true false false 0.33092167108276527 0 1.99502394038673243258657916992 ["ModularForm/GL2/Q/holomorphic/1805/1/o/a/1029/1"] "2-1805-95.82-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.82" [] [[0.0, 0.0]] 0 true true false false -0.14355577018394375 0 0.52441140954466917392522906680 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/1317/1"] "2-1805-95.83-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.83" [] [[0.0, 0.0]] 0 true true false false 0.0203508075251149 0 1.28268344158568179228032519218 ["ModularForm/GL2/Q/holomorphic/1805/1/m/a/653/1"] "2-1805-95.84-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.84" [] [[0.0, 0.0]] 0 true true false false -0.36720997736447175 0 0.63738001368318639160124004468 ["ModularForm/GL2/Q/holomorphic/1805/1/h/a/654/1"] "2-1805-95.84-c0-0-1" 0.9491113463669947 0.9008123478025694 2 1805 "95.84" [] [[0.0, 0.0]] 0 true true false false 0.13279002263552828 0 1.34575692393145751466489546728 ["ModularForm/GL2/Q/holomorphic/1805/1/h/b/654/2"] "2-1805-95.84-c0-0-2" 0.9491113463669947 0.9008123478025694 2 1805 "95.84" [] [[0.0, 0.0]] 0 true true false false 0.13279002263552828 0 1.86904023238677044896944476337 ["ModularForm/GL2/Q/holomorphic/1805/1/h/b/654/1"] "2-1805-95.87-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.87" [] [[0.0, 0.0]] 0 true true false false -0.0203508075251149 0 1.23740987020273788573557037796 ["ModularForm/GL2/Q/holomorphic/1805/1/m/a/1512/1"] "2-1805-95.89-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.89" [] [[0.0, 0.0]] 0 true true false false -0.33092167108276527 0 0.76050564590831032545194604159 ["ModularForm/GL2/Q/holomorphic/1805/1/o/a/849/1"] "2-1805-95.89-c0-0-1" 0.9491113463669947 0.9008123478025694 2 1805 "95.89" [] [[0.0, 0.0]] 0 true true false false 0.1690783289172348 0 1.16145643419303792214584905934 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/849/1"] "2-1805-95.89-c0-0-2" 0.9491113463669947 0.9008123478025694 2 1805 "95.89" [] [[0.0, 0.0]] 0 true true false false 0.1690783289172348 0 2.07527952267958783887988050643 ["ModularForm/GL2/Q/holomorphic/1805/1/o/b/849/2"] "2-1805-95.92-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.92" [] [[0.0, 0.0]] 0 true true false false -0.03265242099083963 0 1.39083430413202575321031520525 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/1137/1"] "2-1805-95.93-c0-0-0" 0.9491113463669947 0.9008123478025694 2 1805 "95.93" [] [[0.0, 0.0]] 0 true true false false 0.23068018847907307 0 1.92277811208098632257182543360 ["ModularForm/GL2/Q/holomorphic/1805/1/r/a/1328/1"] # 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).