LMFDB - L-function 2-39-39.38-c12-0-41

Dirichlet series

L(s) = 1

− 729·3^-s − 4.09e3·4^-s − 1.78e5i·7^-s + 5.31e5·9^-s + 2.98e6·12^-s + (4.69e6 + 1.10e6i)13^-s + 1.67e7·16^-s − 9.23e7i·19^-s + 1.30e8i·21^-s − 2.44e8·25^-s − 3.87e8·27^-s + 7.30e8i·28^-s − 1.69e9i·31^-s − 2.17e9·36^-s + 4.28e9i·37^-s + ⋯

L(s) = 1

− 0.999·3^-s − 4^-s − 1.51i·7^-s + 9^-s + 0.999·12^-s + (0.973 + 0.228i)13^-s + 16^-s − 1.96i·19^-s + 1.51i·21^-s − 25^-s − 27^-s + 1.51i·28^-s − 1.90i·31^-s − 36^-s + 1.66i·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(13-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$39$ = $3 \cdot 13$
Sign:	$-0.973 - 0.228i$
Analytic conductor:	$35.6457$
Root analytic conductor:	$5.97040$
Motivic weight:	$12$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{39} (38, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 39,\ (\ :6),\ -0.973 - 0.228i)$

Particular Values

$L(\frac{13}{2})$	$\approx$	$0.3940981971$
$L(\frac12)$	$\approx$	$0.3940981971$
$L(7)$		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 + 729T $
	13	$ 1 + (-4.69e6 - 1.10e6i)T $
good	2	$ 1 + 4.09e3T^{2} $
	5	$ 1 + 2.44e8T^{2} $
	7	$ 1 + 1.78e5iT - 1.38e10T^{2} $
	11	$ 1 + 3.13e12T^{2} $
	17	$ 1 - 5.82e14T^{2} $
	19	$ 1 + 9.23e7iT - 2.21e15T^{2} $
	23	$ 1 - 2.19e16T^{2} $
	29	$ 1 - 3.53e17T^{2} $
	31	$ 1 + 1.69e9iT - 7.87e17T^{2} $
	37	$ 1 - 4.28e9iT - 6.58e18T^{2} $
	41	$ 1 + 2.25e19T^{2} $
	43	$ 1 + 2.35e8T + 3.99e19T^{2} $
	47	$ 1 + 1.16e20T^{2} $
	53	$ 1 - 4.91e20T^{2} $
	59	$ 1 + 1.77e21T^{2} $
	61	$ 1 + 7.40e10T + 2.65e21T^{2} $
	67	$ 1 - 9.96e10iT - 8.18e21T^{2} $
	71	$ 1 + 1.64e22T^{2} $
	73	$ 1 + 2.84e11iT - 2.29e22T^{2} $
	79	$ 1 - 4.44e11T + 5.90e22T^{2} $
	83	$ 1 + 1.06e23T^{2} $
	89	$ 1 + 2.46e23T^{2} $
	97	$ 1 + 1.02e11iT - 6.93e23T^{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

−13.34339643426921121778730174488, −11.54838844429657029517114710809, −10.53569103467928074855108174670, −9.409839684916291299148386666874, −7.70281839259229026435159895160, −6.35678768398184177010720867411, −4.77770637456620790965374592487, −3.89596297058787303342182521529, −1.06182425446566030401999970022, −0.17723438030748197923779599392, 1.50779360045767137742710628023, 3.75679987417177847333319871792, 5.33671094507001116652798607721, 6.02609216567726439380902690433, 8.125883986680022357579373023222, 9.304232954931163860701137828954, 10.56025949324996381139666896501, 12.06129481365872327388717970652, 12.67963841100173459334283810991, 14.12639071869767478726388778054

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 $\F_{q}$

\(L(\frac{13}{2})\)	\(\approx\)	\(0.3940981971\)
\(L(\frac12)\)	\(\approx\)	\(0.3940981971\)
\(L(7)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more