LMFDB - L-function 2-4004-13.12-c1-0-30

Dirichlet series

L(s) = 1

− 2.10·3^-s + 3.39i·5^-s − i·7^-s + 1.44·9^-s + i·11^-s + (−3.53 − 0.733i)13^-s − 7.15i·15^-s + 6.05·17^-s + 1.08i·19^-s + 2.10i·21^-s − 6.38·23^-s − 6.49·25^-s + 3.26·27^-s − 1.88·29^-s − 5.69i·31^-s + ⋯

L(s) = 1

− 1.21·3^-s + 1.51i·5^-s − 0.377i·7^-s + 0.483·9^-s + 0.301i·11^-s + (−0.979 − 0.203i)13^-s − 1.84i·15^-s + 1.46·17^-s + 0.248i·19^-s + 0.460i·21^-s − 1.33·23^-s − 1.29·25^-s + 0.629·27^-s − 0.349·29^-s − 1.02i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4004$ = $2^{2} \cdot 7 \cdot 11 \cdot 13$
Sign:	$0.979 + 0.203i$
Analytic conductor:	$31.9721$
Root analytic conductor:	$5.65438$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4004} (2157, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4004,\ (\ :1/2),\ 0.979 + 0.203i)$

Particular Values

$L(1)$	$\approx$	$0.7471388668$
$L(\frac12)$	$\approx$	$0.7471388668$
$L(\frac{3}{2})$		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 $
	7	$ 1 + iT $
	11	$ 1 - iT $
	13	$ 1 + (3.53 + 0.733i)T $
good	3	$ 1 + 2.10T + 3T^{2} $
	5	$ 1 - 3.39iT - 5T^{2} $
	17	$ 1 - 6.05T + 17T^{2} $
	19	$ 1 - 1.08iT - 19T^{2} $
	23	$ 1 + 6.38T + 23T^{2} $
	29	$ 1 + 1.88T + 29T^{2} $
	31	$ 1 + 5.69iT - 31T^{2} $
	37	$ 1 + 5.35iT - 37T^{2} $
	41	$ 1 - 1.69iT - 41T^{2} $
	43	$ 1 - 8.15T + 43T^{2} $
	47	$ 1 + 2.26iT - 47T^{2} $
	53	$ 1 + 12.6T + 53T^{2} $
	59	$ 1 + 9.70iT - 59T^{2} $
	61	$ 1 - 1.04T + 61T^{2} $
	67	$ 1 - 6.67iT - 67T^{2} $
	71	$ 1 + 13.9iT - 71T^{2} $
	73	$ 1 - 8.29iT - 73T^{2} $
	79	$ 1 + 1.23T + 79T^{2} $
	83	$ 1 + 10.9iT - 83T^{2} $
	89	$ 1 + 2.96iT - 89T^{2} $
	97	$ 1 - 0.535iT - 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

−8.032734686725994307144525246346, −7.52686152108456136098754717414, −6.95157735481327067019796477424, −6.05115127352896601034778121859, −5.75801290463111984551423980351, −4.75578104678587771817836991813, −3.81081849077421552314091758029, −2.96132683623164977822653275070, −1.96307278846603756022735909441, −0.38832585510219955961982776614, 0.71379219390788660961122750014, 1.64090534490189327338291664847, 2.99379945897911601694719969659, 4.25926316478722106031504885132, 4.88217648496367531563984270090, 5.54966669417862155789672979242, 5.88747927801569445700644510659, 6.90823382835388317618175473035, 7.87057975434021886676352765438, 8.398203047086051916730025204531

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7471388668\)
\(L(\frac12)\)	\(\approx\)	\(0.7471388668\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more