LMFDB - L-function 2-675-5.4-c3-0-3

Dirichlet series

L(s) = 1

− 0.258i·2^-s + 7.93·4^-s + 14.5i·7^-s − 4.12i·8^-s − 49.2·11^-s − 72.1i·13^-s + 3.75·14^-s + 62.3·16^-s + 118. i·17^-s − 123.·19^-s + 12.7i·22^-s + 91.4i·23^-s − 18.6·26^-s + 115. i·28^-s − 174.·29^-s + ⋯

L(s) = 1

− 0.0914i·2^-s + 0.991·4^-s + 0.783i·7^-s − 0.182i·8^-s − 1.35·11^-s − 1.53i·13^-s + 0.0716·14^-s + 0.974·16^-s + 1.68i·17^-s − 1.48·19^-s + 0.123i·22^-s + 0.829i·23^-s − 0.140·26^-s + 0.777i·28^-s − 1.11·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$-0.894 - 0.447i$
Analytic conductor:	$39.8262$
Root analytic conductor:	$6.31080$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :3/2),\ -0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$0.6212757489$
$L(\frac12)$	$\approx$	$0.6212757489$
$L(\frac{5}{2})$		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 $
good	2	$ 1 + 0.258iT - 8T^{2} $
	7	$ 1 - 14.5iT - 343T^{2} $
	11	$ 1 + 49.2T + 1.33e3T^{2} $
	13	$ 1 + 72.1iT - 2.19e3T^{2} $
	17	$ 1 - 118. iT - 4.91e3T^{2} $
	19	$ 1 + 123.T + 6.85e3T^{2} $
	23	$ 1 - 91.4iT - 1.21e4T^{2} $
	29	$ 1 + 174.T + 2.43e4T^{2} $
	31	$ 1 + 46.2T + 2.97e4T^{2} $
	37	$ 1 - 154. iT - 5.06e4T^{2} $
	41	$ 1 + 364.T + 6.89e4T^{2} $
	43	$ 1 + 125. iT - 7.95e4T^{2} $
	47	$ 1 + 221. iT - 1.03e5T^{2} $
	53	$ 1 - 13.6iT - 1.48e5T^{2} $
	59	$ 1 + 239.T + 2.05e5T^{2} $
	61	$ 1 + 54.5T + 2.26e5T^{2} $
	67	$ 1 + 76.0iT - 3.00e5T^{2} $
	71	$ 1 - 728.T + 3.57e5T^{2} $
	73	$ 1 - 501. iT - 3.89e5T^{2} $
	79	$ 1 + 397.T + 4.93e5T^{2} $
	83	$ 1 - 1.36e3iT - 5.71e5T^{2} $
	89	$ 1 + 1.46e3T + 7.04e5T^{2} $
	97	$ 1 - 335. iT - 9.12e5T^{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

−10.61599319341061244287706251642, −9.896071298435939174177089975954, −8.352751852255948758536438133900, −8.090380598096620432930864017342, −6.93090089144567606082046519886, −5.81094373368700193800061468753, −5.41508969478809445375395560795, −3.68637661612565338784043519880, −2.66642465873837629777824048395, −1.76175452185414406853943400895, 0.14747141585450577505130855490, 1.87323775722988579917648050132, 2.77327484339063949094144368395, 4.15638810995570497326385646030, 5.15606432488061929765034508022, 6.38442931848689909853872108103, 7.09327112808991955639842608958, 7.72651242905089997586341670834, 8.839167851644644163532901635382, 9.883982318979785183456814034251

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.6212757489\)
\(L(\frac12)\)	\(\approx\)	\(0.6212757489\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more