LMFDB - L-function 2-7200-24.11-c1-0-38

Dirichlet series

L(s) = 1

− 3.02i·7^-s − 3.62i·11^-s + 1.69i·13^-s + 6.60i·17^-s + 5.12·19^-s + 6.67·23^-s − 6.82·29^-s + 1.73i·31^-s − 0.371i·37^-s + 5.83i·41^-s + 5.24·43^-s − 0.525·47^-s − 2.12·49^-s + 10.0·53^-s + 4.86i·59^-s + ⋯

L(s) = 1

− 1.14i·7^-s − 1.09i·11^-s + 0.470i·13^-s + 1.60i·17^-s + 1.17·19^-s + 1.39·23^-s − 1.26·29^-s + 0.311i·31^-s − 0.0611i·37^-s + 0.910i·41^-s + 0.799·43^-s − 0.0767·47^-s − 0.303·49^-s + 1.37·53^-s + 0.633i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.971 + 0.238i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (4751, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ 0.971 + 0.238i)$

Particular Values

$L(1)$	$\approx$	$2.093364131$
$L(\frac12)$	$\approx$	$2.093364131$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	7	$ 1 + 3.02iT - 7T^{2} $
	11	$ 1 + 3.62iT - 11T^{2} $
	13	$ 1 - 1.69iT - 13T^{2} $
	17	$ 1 - 6.60iT - 17T^{2} $
	19	$ 1 - 5.12T + 19T^{2} $
	23	$ 1 - 6.67T + 23T^{2} $
	29	$ 1 + 6.82T + 29T^{2} $
	31	$ 1 - 1.73iT - 31T^{2} $
	37	$ 1 + 0.371iT - 37T^{2} $
	41	$ 1 - 5.83iT - 41T^{2} $
	43	$ 1 - 5.24T + 43T^{2} $
	47	$ 1 + 0.525T + 47T^{2} $
	53	$ 1 - 10.0T + 53T^{2} $
	59	$ 1 - 4.86iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 13.4T + 67T^{2} $
	71	$ 1 - 2.45T + 71T^{2} $
	73	$ 1 - 14.5T + 73T^{2} $
	79	$ 1 - 14.1iT - 79T^{2} $
	83	$ 1 + 5.79iT - 83T^{2} $
	89	$ 1 - 10.2iT - 89T^{2} $
	97	$ 1 - 9.33T + 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

−7.79017634489783346794752518488, −7.26184694992122962650585802697, −6.55151822760098234038606701827, −5.80414024959568280706235308267, −5.13383724319473476367566334852, −4.09599994847424995526508030875, −3.66488091943659914569990885961, −2.83599061316187158867171640810, −1.52301826448816342839002860819, −0.802013012685434959335636121917, 0.72104077556235776264695721952, 1.99836495583581967761633695524, 2.70208251019265600033736130751, 3.42170632005597255810383753548, 4.55417054964037749350377950533, 5.30157593873645526087194242323, 5.55337088272105519513926167190, 6.66384380087283948076258570738, 7.41434248636300756092298655983, 7.66648064299912809419144780229

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.093364131\)
\(L(\frac12)\)	\(\approx\)	\(2.093364131\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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