LMFDB - L-function 2-40e2-8.5-c1-0-2

Dirichlet series

L(s) = 1

− 3.44i·3^-s − 8.89·9^-s + 5.44i·11^-s − 1.89·17^-s + 6.34i·19^-s + 20.3i·27^-s + 18.7·33^-s − 6.79·41^-s + 10i·43^-s − 7·49^-s + 6.55i·51^-s + 21.8·57^-s − 6i·59^-s − 0.348i·67^-s − 15.6·73^-s + ⋯

L(s) = 1

− 1.99i·3^-s − 2.96·9^-s + 1.64i·11^-s − 0.460·17^-s + 1.45i·19^-s + 3.91i·27^-s + 3.27·33^-s − 1.06·41^-s + 1.52i·43^-s − 49^-s + 0.917i·51^-s + 2.90·57^-s − 0.781i·59^-s − 0.0425i·67^-s − 1.83·73^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1600$ = $2^{6} \cdot 5^{2}$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$12.7760$
Root analytic conductor:	$3.57436$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1600} (801, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1600,\ (\ :1/2),\ 0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.6653378516$
$L(\frac12)$	$\approx$	$0.6653378516$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 $
good	3	$ 1 + 3.44iT - 3T^{2} $
	7	$ 1 + 7T^{2} $
	11	$ 1 - 5.44iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 + 1.89T + 17T^{2} $
	19	$ 1 - 6.34iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 + 6.79T + 41T^{2} $
	43	$ 1 - 10iT - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 6iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 0.348iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 15.6T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 6.55iT - 83T^{2} $
	89	$ 1 + 4.10T + 89T^{2} $
	97	$ 1 - 10T + 97T^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−9.377720568316732100557639992082, −8.363880547781772986585580708868, −7.80963004480245790585534356018, −7.11304003132228460198906431917, −6.49658528291289484565504829893, −5.72124185109365549414477044976, −4.64275188382580298493315496524, −3.19480130576218244954632182829, −2.06475766542034979776729449930, −1.47091777749652975528380604914, 0.24896566273587884968868717262, 2.70193516356362930890614125162, 3.40249798559737535107142791927, 4.25126532216924503449345604513, 5.10704055878021143062327891355, 5.73500833370211065876594668818, 6.67634766185253017076038313156, 8.139718181697095781098210746919, 8.855239521410701652254725544393, 9.158761557761579279104576565086

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6653378516\)
\(L(\frac12)\)	\(\approx\)	\(0.6653378516\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more