LMFDB - L-function 2-2475-165.164-c1-0-17

Dirichlet series

L(s) = 1

− 2.52i·2^-s − 4.36·4^-s − 2.99·7^-s + 5.95i·8^-s + (−3.25 − 0.639i)11^-s + 4.04·13^-s + 7.54i·14^-s + 6.30·16^-s + 0.242i·17^-s + 5.43i·19^-s + (−1.61 + 8.20i)22^-s − 2.10·23^-s − 10.1i·26^-s + 13.0·28^-s − 7.68·29^-s + ⋯

L(s) = 1

− 1.78i·2^-s − 2.18·4^-s − 1.13·7^-s + 2.10i·8^-s + (−0.981 − 0.192i)11^-s + 1.12·13^-s + 2.01i·14^-s + 1.57·16^-s + 0.0588i·17^-s + 1.24i·19^-s + (−0.343 + 1.75i)22^-s − 0.438·23^-s − 1.99i·26^-s + 2.46·28^-s − 1.42·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0420 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0420 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2475$ = $3^{2} \cdot 5^{2} \cdot 11$
Sign:	$-0.0420 + 0.999i$
Analytic conductor:	$19.7629$
Root analytic conductor:	$4.44555$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2475} (2474, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2475,\ (\ :1/2),\ -0.0420 + 0.999i)$

Particular Values

$L(1)$	$\approx$	$0.9903041845$
$L(\frac12)$	$\approx$	$0.9903041845$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
	11	$ 1 + (3.25 + 0.639i)T $
good	2	$ 1 + 2.52iT - 2T^{2} $
	7	$ 1 + 2.99T + 7T^{2} $
	13	$ 1 - 4.04T + 13T^{2} $
	17	$ 1 - 0.242iT - 17T^{2} $
	19	$ 1 - 5.43iT - 19T^{2} $
	23	$ 1 + 2.10T + 23T^{2} $
	29	$ 1 + 7.68T + 29T^{2} $
	31	$ 1 - 7.30T + 31T^{2} $
	37	$ 1 + 0.554iT - 37T^{2} $
	41	$ 1 - 0.219T + 41T^{2} $
	43	$ 1 - 5.16T + 43T^{2} $
	47	$ 1 - 10.2T + 47T^{2} $
	53	$ 1 - 11.6T + 53T^{2} $
	59	$ 1 + 10.8iT - 59T^{2} $
	61	$ 1 - 2.11iT - 61T^{2} $
	67	$ 1 - 13.8iT - 67T^{2} $
	71	$ 1 - 3.51iT - 71T^{2} $
	73	$ 1 + 7.64T + 73T^{2} $
	79	$ 1 + 14.5iT - 79T^{2} $
	83	$ 1 - 7.78iT - 83T^{2} $
	89	$ 1 + 11.6iT - 89T^{2} $
	97	$ 1 + 3.13iT - 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

−8.934769745785391461574963357772, −8.374205775666209788588238127269, −7.38554805549844196032596511908, −6.08749911692271657056339275224, −5.55402761652381274058679455546, −4.24241416427757896205335080729, −3.64267148173456483178315848122, −2.91510432276981523122778655375, −1.99457051998969423056943482956, −0.75110424222906439405596423326, 0.51202733468115979878144378599, 2.63308375150326618343003600849, 3.77246273131492025065087110284, 4.59852875324656343411678274030, 5.61025088967344931237940371248, 6.03338953647546354482814580716, 6.86909522029306035285465875643, 7.40686147695129341331431158769, 8.220789298451555252407532924693, 8.937055647292413637215165866845

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9903041845\)
\(L(\frac12)\)	\(\approx\)	\(0.9903041845\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more