LMFDB - L-function 2-320-40.19-c2-0-16

Dirichlet series

L(s) = 1

− 1.90i·3^-s + (4.37 − 2.41i)5^-s − 3.95·7^-s + 5.35·9^-s + 6.18·11^-s + 4.94·13^-s + (−4.60 − 8.35i)15^-s − 22.4i·17^-s + 10.3·19^-s + 7.54i·21^-s − 39.6·23^-s + (13.3 − 21.1i)25^-s − 27.4i·27^-s + 30.4i·29^-s − 24.7i·31^-s + ⋯

L(s) = 1

− 0.636i·3^-s + (0.875 − 0.482i)5^-s − 0.565·7^-s + 0.595·9^-s + 0.562·11^-s + 0.380·13^-s + (−0.306 − 0.557i)15^-s − 1.31i·17^-s + 0.546·19^-s + 0.359i·21^-s − 1.72·23^-s + (0.534 − 0.845i)25^-s − 1.01i·27^-s + 1.04i·29^-s − 0.797i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$320$ = $2^{6} \cdot 5$
Sign:	$0.239 + 0.970i$
Analytic conductor:	$8.71936$
Root analytic conductor:	$2.95285$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{320} (159, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 320,\ (\ :1),\ 0.239 + 0.970i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.49403 - 1.17039i$
$L(\frac12)$	$\approx$	$1.49403 - 1.17039i$
$L(2)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 + (-4.37 + 2.41i)T $
good	3	$ 1 + 1.90iT - 9T^{2} $
	7	$ 1 + 3.95T + 49T^{2} $
	11	$ 1 - 6.18T + 121T^{2} $
	13	$ 1 - 4.94T + 169T^{2} $
	17	$ 1 + 22.4iT - 289T^{2} $
	19	$ 1 - 10.3T + 361T^{2} $
	23	$ 1 + 39.6T + 529T^{2} $
	29	$ 1 - 30.4iT - 841T^{2} $
	31	$ 1 + 24.7iT - 961T^{2} $
	37	$ 1 - 24.0T + 1.36e3T^{2} $
	41	$ 1 - 31.0T + 1.68e3T^{2} $
	43	$ 1 + 52.2iT - 1.84e3T^{2} $
	47	$ 1 + 13.1T + 2.20e3T^{2} $
	53	$ 1 + 17.9T + 2.80e3T^{2} $
	59	$ 1 - 104.T + 3.48e3T^{2} $
	61	$ 1 + 57.2iT - 3.72e3T^{2} $
	67	$ 1 - 99.3iT - 4.48e3T^{2} $
	71	$ 1 - 16.7iT - 5.04e3T^{2} $
	73	$ 1 - 96.3iT - 5.32e3T^{2} $
	79	$ 1 - 139. iT - 6.24e3T^{2} $
	83	$ 1 - 91.7iT - 6.88e3T^{2} $
	89	$ 1 + 94.7T + 7.92e3T^{2} $
	97	$ 1 - 143. 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

−11.39169000744669529397874824063, −9.924819705991093481772971807127, −9.575805366183055018169957418234, −8.399819006517541598335274815324, −7.20544362373926130149855095656, −6.39394173255594105987100149840, −5.39331186150940331874109481781, −4.00257188355793199318921905479, −2.33410637529702809213659803348, −0.996397206492402473603460706553, 1.72484140926247125947457229090, 3.36428275495060338761996409717, 4.37203039614888366789362041971, 5.90350770789946638549876390086, 6.47677311602882811716121896101, 7.80714900360265350789112052640, 9.110900088472421280791214036462, 9.914277567354416626984699523173, 10.36433090480890156265717423446, 11.45573478016815843790666173427

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.49403 - 1.17039i\)
\(L(\frac12)\)	\(\approx\)	\(1.49403 - 1.17039i\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more