LMFDB - L-function 2-2205-105.104-c1-0-49

Dirichlet series

L(s) = 1

− 1.91·2^-s + 1.66·4^-s + (0.630 − 2.14i)5^-s + 0.650·8^-s + (−1.20 + 4.10i)10^-s + 3.22i·11^-s − 4.86·13^-s − 4.56·16^-s + 0.729i·17^-s + 7.87i·19^-s + (1.04 − 3.56i)20^-s − 6.17i·22^-s + 4.86·23^-s + (−4.20 − 2.70i)25^-s + 9.30·26^-s + ⋯

L(s) = 1

− 1.35·2^-s + 0.830·4^-s + (0.282 − 0.959i)5^-s + 0.229·8^-s + (−0.381 + 1.29i)10^-s + 0.973i·11^-s − 1.34·13^-s − 1.14·16^-s + 0.176i·17^-s + 1.80i·19^-s + (0.234 − 0.796i)20^-s − 1.31i·22^-s + 1.01·23^-s + (−0.840 − 0.541i)25^-s + 1.82·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2205$ = $3^{2} \cdot 5 \cdot 7^{2}$
Sign:	$-0.503 + 0.864i$
Analytic conductor:	$17.6070$
Root analytic conductor:	$4.19607$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2205} (2204, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2205,\ (\ :1/2),\ -0.503 + 0.864i)$

Particular Values

$L(1)$	$\approx$	$0.4518912142$
$L(\frac12)$	$\approx$	$0.4518912142$
$L(\frac{3}{2})$		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 + (-0.630 + 2.14i)T $
	7	$ 1 $
good	2	$ 1 + 1.91T + 2T^{2} $
	11	$ 1 - 3.22iT - 11T^{2} $
	13	$ 1 + 4.86T + 13T^{2} $
	17	$ 1 - 0.729iT - 17T^{2} $
	19	$ 1 - 7.87iT - 19T^{2} $
	23	$ 1 - 4.86T + 23T^{2} $
	29	$ 1 + 7.75iT - 29T^{2} $
	31	$ 1 + 1.42iT - 31T^{2} $
	37	$ 1 + 3.16iT - 37T^{2} $
	41	$ 1 - 1.40T + 41T^{2} $
	43	$ 1 + 6.42iT - 43T^{2} $
	47	$ 1 + 4.87iT - 47T^{2} $
	53	$ 1 + 1.52T + 53T^{2} $
	59	$ 1 - 6.30T + 59T^{2} $
	61	$ 1 - 2.37iT - 61T^{2} $
	67	$ 1 + 11.1iT - 67T^{2} $
	71	$ 1 + 10.1iT - 71T^{2} $
	73	$ 1 + 13.8T + 73T^{2} $
	79	$ 1 - 3.98T + 79T^{2} $
	83	$ 1 + 4.19iT - 83T^{2} $
	89	$ 1 - 11.2T + 89T^{2} $
	97	$ 1 - 2.21T + 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

−8.975440756093440117197785002854, −7.972920434652579505979007683992, −7.67390447165787916047382263114, −6.75460000883440190848346730833, −5.63727655744098257178611528914, −4.80007012809996373954582713972, −4.02924850822797843943889754896, −2.30341055079675923556951219038, −1.60770790314869062323947161677, −0.28675212481891010545719554584, 1.07439242444228427788670503344, 2.50761264502075596904587693286, 3.06048493057376662486282336148, 4.59417046458992979896683525607, 5.42117789999812930943487519314, 6.75672356152567653152417407040, 6.99395499334590375808894758882, 7.80079017584421746164708149868, 8.729062450009579659252827248834, 9.286140385325512959763484777467

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4518912142\)
\(L(\frac12)\)	\(\approx\)	\(0.4518912142\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more