LMFDB - L-function 2-3744-104.77-c1-0-32

Dirichlet series

L(s) = 1

− 1.81·5^-s + 1.13i·7^-s − 4.40·11^-s + (2.58 + 2.50i)13^-s − 0.701·17^-s − 5.95·19^-s + 4·23^-s − 1.70·25^-s + 5.01i·29^-s − 8.77i·31^-s − 2.06i·35^-s + 3.36·37^-s + 2.94i·43^-s − 1.13i·47^-s + 5.70·49^-s + ⋯

L(s) = 1

− 0.812·5^-s + 0.430i·7^-s − 1.32·11^-s + (0.717 + 0.696i)13^-s − 0.170·17^-s − 1.36·19^-s + 0.834·23^-s − 0.340·25^-s + 0.932i·29^-s − 1.57i·31^-s − 0.349i·35^-s + 0.552·37^-s + 0.449i·43^-s − 0.166i·47^-s + 0.814·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$0.376 + 0.926i$
Analytic conductor:	$29.8959$
Root analytic conductor:	$5.46772$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3744} (1585, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3744,\ (\ :1/2),\ 0.376 + 0.926i)$

Particular Values

$L(1)$	$\approx$	$0.8070926561$
$L(\frac12)$	$\approx$	$0.8070926561$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	13	$ 1 + (-2.58 - 2.50i)T $
good	5	$ 1 + 1.81T + 5T^{2} $
	7	$ 1 - 1.13iT - 7T^{2} $
	11	$ 1 + 4.40T + 11T^{2} $
	17	$ 1 + 0.701T + 17T^{2} $
	19	$ 1 + 5.95T + 19T^{2} $
	23	$ 1 - 4T + 23T^{2} $
	29	$ 1 - 5.01iT - 29T^{2} $
	31	$ 1 + 8.77iT - 31T^{2} $
	37	$ 1 - 3.36T + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 - 2.94iT - 43T^{2} $
	47	$ 1 + 1.13iT - 47T^{2} $
	53	$ 1 + 11.7iT - 53T^{2} $
	59	$ 1 + 5.95T + 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 7.49T + 67T^{2} $
	71	$ 1 - 11.8iT - 71T^{2} $
	73	$ 1 + 8.43iT - 73T^{2} $
	79	$ 1 - 10.8T + 79T^{2} $
	83	$ 1 + 2.85T + 83T^{2} $
	89	$ 1 + 8.43iT - 89T^{2} $
	97	$ 1 + 12.9iT - 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

−8.329095746249053540874134743149, −7.77333629431195889291010257249, −6.93743556473254156037840056046, −6.16553495738354301520355761844, −5.35106323460766488413911700141, −4.48526562704802474637878631910, −3.80566841099958635822396118795, −2.78126537728873645182815283330, −1.91834077938564831346231556786, −0.30798957590072330114506354139, 0.842153857330432863321824175869, 2.31576855882034984737027359686, 3.21571498197371858251622837768, 4.04768703895969619097054012251, 4.78576221541207696718156158165, 5.64617314430194156708487779982, 6.47746897804086047980810978012, 7.33395572141926570656038717888, 7.974118313919883296557890336560, 8.421191425847326258610857647264

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8070926561\)
\(L(\frac12)\)	\(\approx\)	\(0.8070926561\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more