LMFDB - L-function 2-855-5.4-c1-0-30

Dirichlet series

L(s) = 1

− 0.906i·2^-s + 1.17·4^-s + (0.370 + 2.20i)5^-s − 2.59i·7^-s − 2.88i·8^-s + (1.99 − 0.336i)10^-s − 0.741·11^-s − 3.78i·13^-s − 2.35·14^-s − 0.258·16^-s − 3.16i·17^-s + 19^-s + (0.436 + 2.59i)20^-s + 0.672i·22^-s − 0.570i·23^-s + ⋯

L(s) = 1

− 0.641i·2^-s + 0.588·4^-s + (0.165 + 0.986i)5^-s − 0.981i·7^-s − 1.01i·8^-s + (0.632 − 0.106i)10^-s − 0.223·11^-s − 1.05i·13^-s − 0.629·14^-s − 0.0647·16^-s − 0.768i·17^-s + 0.229·19^-s + (0.0975 + 0.580i)20^-s + 0.143i·22^-s − 0.119i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$855$ = $3^{2} \cdot 5 \cdot 19$
Sign:	$0.165 + 0.986i$
Analytic conductor:	$6.82720$
Root analytic conductor:	$2.61289$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{855} (514, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 855,\ (\ :1/2),\ 0.165 + 0.986i)$

Particular Values

$L(1)$	$\approx$	$1.43716 - 1.21581i$
$L(\frac12)$	$\approx$	$1.43716 - 1.21581i$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 + (-0.370 - 2.20i)T $
	19	$ 1 - T $
good	2	$ 1 + 0.906iT - 2T^{2} $
	7	$ 1 + 2.59iT - 7T^{2} $
	11	$ 1 + 0.741T + 11T^{2} $
	13	$ 1 + 3.78iT - 13T^{2} $
	17	$ 1 + 3.16iT - 17T^{2} $
	23	$ 1 + 0.570iT - 23T^{2} $
	29	$ 1 - 6T + 29T^{2} $
	31	$ 1 - 5.83T + 31T^{2} $
	37	$ 1 - 1.40iT - 37T^{2} $
	41	$ 1 - 3.83T + 41T^{2} $
	43	$ 1 + 2.59iT - 43T^{2} $
	47	$ 1 - 5.08iT - 47T^{2} $
	53	$ 1 - 0.160iT - 53T^{2} $
	59	$ 1 - 8.35T + 59T^{2} $
	61	$ 1 + 8.57T + 61T^{2} $
	67	$ 1 - 14.8iT - 67T^{2} $
	71	$ 1 + 3.64T + 71T^{2} $
	73	$ 1 + 10.8iT - 73T^{2} $
	79	$ 1 + 1.83T + 79T^{2} $
	83	$ 1 - 4.19iT - 83T^{2} $
	89	$ 1 + 16.9T + 89T^{2} $
	97	$ 1 - 3.78iT - 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−10.32468240158380320972306879611, −9.603501358918820012999657483820, −8.094663516375591439809703847252, −7.30763250343128350611339042101, −6.70846533133356913665145977568, −5.72675792090165696003312939635, −4.31798559317634937561398688010, −3.17644940240857250388479928074, −2.58514641659851077324678392742, −0.961246311443921929651030259647, 1.63373839324785596726698187873, 2.67431927381163926755411113721, 4.32289491246926012575726101773, 5.29521076006401585573314191413, 6.02149632613426477450951275997, 6.78761352205866707793526320130, 7.969544884928370259141898104442, 8.540503832087814769025826280019, 9.281037133292102684033768338719, 10.27103432802632473901376876810

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.43716 - 1.21581i\)
\(L(\frac12)\)	\(\approx\)	\(1.43716 - 1.21581i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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