LMFDB - L-function 2-3744-8.5-c1-0-51

Dirichlet series

L(s) = 1

− 2.52i·5^-s − 2.79·7^-s + 5.31i·11^-s − i·13^-s + 4.35·17^-s − 6.02i·19^-s + 0.568·23^-s − 1.35·25^-s − 2.68i·29^-s − 1.90·31^-s + 7.04i·35^-s − 9.60i·37^-s − 11.2·41^-s + 9.48i·43^-s + 3.95·47^-s + ⋯

L(s) = 1

− 1.12i·5^-s − 1.05·7^-s + 1.60i·11^-s − 0.277i·13^-s + 1.05·17^-s − 1.38i·19^-s + 0.118·23^-s − 0.270·25^-s − 0.499i·29^-s − 0.342·31^-s + 1.19i·35^-s − 1.57i·37^-s − 1.75·41^-s + 1.44i·43^-s + 0.577·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$-0.987 + 0.156i$
Analytic conductor:	$29.8959$
Root analytic conductor:	$5.46772$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3744} (1873, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3744,\ (\ :1/2),\ -0.987 + 0.156i)$

Particular Values

$L(1)$	$\approx$	$0.5422126905$
$L(\frac12)$	$\approx$	$0.5422126905$
$L(\frac{3}{2})$		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 + iT $
good	5	$ 1 + 2.52iT - 5T^{2} $
	7	$ 1 + 2.79T + 7T^{2} $
	11	$ 1 - 5.31iT - 11T^{2} $
	17	$ 1 - 4.35T + 17T^{2} $
	19	$ 1 + 6.02iT - 19T^{2} $
	23	$ 1 - 0.568T + 23T^{2} $
	29	$ 1 + 2.68iT - 29T^{2} $
	31	$ 1 + 1.90T + 31T^{2} $
	37	$ 1 + 9.60iT - 37T^{2} $
	41	$ 1 + 11.2T + 41T^{2} $
	43	$ 1 - 9.48iT - 43T^{2} $
	47	$ 1 - 3.95T + 47T^{2} $
	53	$ 1 - 8.05iT - 53T^{2} $
	59	$ 1 + 6.19iT - 59T^{2} $
	61	$ 1 + 7.23iT - 61T^{2} $
	67	$ 1 + 10.0iT - 67T^{2} $
	71	$ 1 + 1.63T + 71T^{2} $
	73	$ 1 + 5.17T + 73T^{2} $
	79	$ 1 + 9.17T + 79T^{2} $
	83	$ 1 + 4.61iT - 83T^{2} $
	89	$ 1 + 11.4T + 89T^{2} $
	97	$ 1 + 10.2T + 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

−8.165480099688348503597171617494, −7.38466173057953135671159673471, −6.79212688395893359189991459098, −5.85686445894700272126535210597, −5.01805962004054860256934829372, −4.50216346798755621775090876753, −3.49064573154516254471347525658, −2.52430395943699002994995028015, −1.37880267823240858185578209868, −0.16435549540283927286501495587, 1.37355942013110525923960914170, 2.86646952246966347485293029557, 3.29064865990239187298910320417, 3.90487179012990344169456360644, 5.45700699615690825701296259470, 5.89421754123455833325203447061, 6.71593136024029837585816703679, 7.16016050223562233481408776038, 8.254356898665954329567215583141, 8.687868304573562412102020361409

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5422126905\)
\(L(\frac12)\)	\(\approx\)	\(0.5422126905\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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