LMFDB - L-function 2-4600-5.4-c1-0-58

Dirichlet series

L(s) = 1

− 0.523i·3^-s − 0.476i·7^-s + 2.72·9^-s − 1.67·11^-s + 2.67i·13^-s − 1.67i·17^-s − 7.92·19^-s − 0.249·21^-s + i·23^-s − 3i·27^-s + 2.20·29^-s + 6.77·31^-s + 0.878i·33^-s − 4i·37^-s + 1.40·39^-s + ⋯

L(s) = 1

− 0.302i·3^-s − 0.179i·7^-s + 0.908·9^-s − 0.505·11^-s + 0.742i·13^-s − 0.406i·17^-s − 1.81·19^-s − 0.0544·21^-s + 0.208i·23^-s − 0.577i·27^-s + 0.408·29^-s + 1.21·31^-s + 0.153i·33^-s − 0.657i·37^-s + 0.224·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4600$ = $2^{3} \cdot 5^{2} \cdot 23$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$36.7311$
Root analytic conductor:	$6.06062$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4600} (4049, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4600,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.737179874$
$L(\frac12)$	$\approx$	$1.737179874$
$L(\frac{3}{2})$		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 $
	23	$ 1 - iT $
good	3	$ 1 + 0.523iT - 3T^{2} $
	7	$ 1 + 0.476iT - 7T^{2} $
	11	$ 1 + 1.67T + 11T^{2} $
	13	$ 1 - 2.67iT - 13T^{2} $
	17	$ 1 + 1.67iT - 17T^{2} $
	19	$ 1 + 7.92T + 19T^{2} $
	29	$ 1 - 2.20T + 29T^{2} $
	31	$ 1 - 6.77T + 31T^{2} $
	37	$ 1 + 4iT - 37T^{2} $
	41	$ 1 - 5.97T + 41T^{2} $
	43	$ 1 + 0.402iT - 43T^{2} $
	47	$ 1 + 1.79iT - 47T^{2} $
	53	$ 1 + 10.8iT - 53T^{2} $
	59	$ 1 + 9.45T + 59T^{2} $
	61	$ 1 - 6.32T + 61T^{2} $
	67	$ 1 + 14.7iT - 67T^{2} $
	71	$ 1 - 9.97T + 71T^{2} $
	73	$ 1 + 6.10iT - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 - 4.49iT - 83T^{2} $
	89	$ 1 - 3.59T + 89T^{2} $
	97	$ 1 - 6.08iT - 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

−8.082784173110893796924619425651, −7.49808677560816723647680434594, −6.62961224463539028750563335382, −6.32048935458583879976922409977, −5.11494389817308811311501058760, −4.42919763886811375532492091587, −3.78630080699195349151673022169, −2.51446714722516750308001293744, −1.83690412465453070576025163005, −0.56559579865520462447109211236, 0.971890300648105959864326566294, 2.20613586778917673374422232963, 2.99221611461249625185523631033, 4.17468367007585049028268617853, 4.51936009156142980338598578972, 5.53421138388121904486451895444, 6.26478752072980560455082236433, 6.95551469414767415003939051119, 7.86940173031690029214670580375, 8.366248885584651390353399409458

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.737179874\)
\(L(\frac12)\)	\(\approx\)	\(1.737179874\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more