LMFDB - L-function 2-2736-57.56-c1-0-21

Dirichlet series

L(s) = 1

− 1.41i·5^-s − 2·7^-s + 1.41i·11^-s − 1.41i·17^-s + (−1 + 4.24i)19^-s + 1.41i·23^-s + 2.99·25^-s + 6·29^-s + 2.82i·35^-s − 8.48i·37^-s + 6·41^-s + 4·43^-s − 7.07i·47^-s − 3·49^-s + 6·53^-s + ⋯

L(s) = 1

− 0.632i·5^-s − 0.755·7^-s + 0.426i·11^-s − 0.342i·17^-s + (−0.229 + 0.973i)19^-s + 0.294i·23^-s + 0.599·25^-s + 1.11·29^-s + 0.478i·35^-s − 1.39i·37^-s + 0.937·41^-s + 0.609·43^-s − 1.03i·47^-s − 0.428·49^-s + 0.824·53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.662 + 0.749i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ 0.662 + 0.749i)$

Particular Values

$L(1)$	$\approx$	$1.495850724$
$L(\frac12)$	$\approx$	$1.495850724$
$L(\frac{3}{2})$		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 $
	19	$ 1 + (1 - 4.24i)T $
good	5	$ 1 + 1.41iT - 5T^{2} $
	7	$ 1 + 2T + 7T^{2} $
	11	$ 1 - 1.41iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 + 1.41iT - 17T^{2} $
	23	$ 1 - 1.41iT - 23T^{2} $
	29	$ 1 - 6T + 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 + 8.48iT - 37T^{2} $
	41	$ 1 - 6T + 41T^{2} $
	43	$ 1 - 4T + 43T^{2} $
	47	$ 1 + 7.07iT - 47T^{2} $
	53	$ 1 - 6T + 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 + 4T + 61T^{2} $
	67	$ 1 + 8.48iT - 67T^{2} $
	71	$ 1 - 12T + 71T^{2} $
	73	$ 1 + 10T + 73T^{2} $
	79	$ 1 + 8.48iT - 79T^{2} $
	83	$ 1 + 15.5iT - 83T^{2} $
	89	$ 1 - 6T + 89T^{2} $
	97	$ 1 - 8.48iT - 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

−8.862011073659789189859265964456, −7.968607039488244870944908177974, −7.23262651496418731159759532417, −6.39907636062997152988100109556, −5.65670085924131563834963298536, −4.78561051033238522822154556347, −3.97679949688989378860562759644, −3.03465511737228209844666263977, −1.92858717990471406488940216022, −0.62343445286279761208255850052, 0.932692176303675529087860756415, 2.56595461323032056866578078774, 3.06769429339843665983831725370, 4.12270903404328261878565353584, 5.00652095774585820789895156975, 6.11957575570286189759501682185, 6.56491759360658942569256961787, 7.27280471617382105116998514886, 8.247565901398626653927898346463, 8.900840881034435884539232854566

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.495850724\)
\(L(\frac12)\)	\(\approx\)	\(1.495850724\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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