LMFDB - L-function 2-675-15.14-c2-0-2

Dirichlet series

L(s) = 1

− 3.69·2^-s + 9.64·4^-s + 9.20i·7^-s − 20.8·8^-s + 15.5i·11^-s + 20.6i·13^-s − 34.0i·14^-s + 38.4·16^-s − 22.4·17^-s − 2.33·19^-s − 57.4i·22^-s − 19.3·23^-s − 76.0i·26^-s + 88.8i·28^-s + 2.92i·29^-s + ⋯

L(s) = 1

− 1.84·2^-s + 2.41·4^-s + 1.31i·7^-s − 2.60·8^-s + 1.41i·11^-s + 1.58i·13^-s − 2.42i·14^-s + 2.40·16^-s − 1.32·17^-s − 0.122·19^-s − 2.61i·22^-s − 0.839·23^-s − 2.92i·26^-s + 3.17i·28^-s + 0.100i·29^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$18.3924$
Root analytic conductor:	$4.28863$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (674, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :1),\ -0.894 + 0.447i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.2629724752$
$L(\frac12)$	$\approx$	$0.2629724752$
$L(2)$		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 + 3.69T + 4T^{2} $
	7	$ 1 - 9.20iT - 49T^{2} $
	11	$ 1 - 15.5iT - 121T^{2} $
	13	$ 1 - 20.6iT - 169T^{2} $
	17	$ 1 + 22.4T + 289T^{2} $
	19	$ 1 + 2.33T + 361T^{2} $
	23	$ 1 + 19.3T + 529T^{2} $
	29	$ 1 - 2.92iT - 841T^{2} $
	31	$ 1 + 28.1T + 961T^{2} $
	37	$ 1 + 65.4iT - 1.36e3T^{2} $
	41	$ 1 - 7.48iT - 1.68e3T^{2} $
	43	$ 1 - 11.9iT - 1.84e3T^{2} $
	47	$ 1 - 50.5T + 2.20e3T^{2} $
	53	$ 1 - 80.8T + 2.80e3T^{2} $
	59	$ 1 + 9.46iT - 3.48e3T^{2} $
	61	$ 1 - 52.7T + 3.72e3T^{2} $
	67	$ 1 + 65.2iT - 4.48e3T^{2} $
	71	$ 1 - 0.942iT - 5.04e3T^{2} $
	73	$ 1 + 12.8iT - 5.32e3T^{2} $
	79	$ 1 + 68.4T + 6.24e3T^{2} $
	83	$ 1 + 38.9T + 6.88e3T^{2} $
	89	$ 1 + 63.7iT - 7.92e3T^{2} $
	97	$ 1 + 186. iT - 9.40e3T^{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

−10.57758584375985371848979023393, −9.561644566335614121456279639096, −9.121629063154706741657186171634, −8.559811190424339770980331616630, −7.35870475778525292342240284308, −6.80064964835937895232872834857, −5.79967735347598620333941412660, −4.28754924005453837787360866059, −2.20470461491553139086681662885, −2.00690884313792020103700423565, 0.19579918079945404992734294801, 1.05078964710140886132073665202, 2.63583948159570125184885029050, 3.83194963886353548978936833711, 5.65200018547171119941555100074, 6.63729093320680882740141943432, 7.44425717374020996253657304315, 8.251457026717938326662797521830, 8.755295031644372425967353114680, 9.929351247795320507943760563156

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 $\F_{q}$
Belyi maps

\(L(\frac{3}{2})\)	\(\approx\)	\(0.2629724752\)
\(L(\frac12)\)	\(\approx\)	\(0.2629724752\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more