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

Dirichlet series

L(s) = 1

− 2.32i·5^-s − 4.34·7^-s − 0.264i·11^-s − 6.38i·13^-s + 2.56i·17^-s + (−2.62 + 3.47i)19^-s − 2.66i·23^-s − 0.426·25^-s − 8.92·29^-s − 1.78i·31^-s + 10.1i·35^-s + 3.08i·37^-s + 7.44·41^-s + 5.05·43^-s + 7.45i·47^-s + ⋯

L(s) = 1

− 1.04i·5^-s − 1.64·7^-s − 0.0797i·11^-s − 1.76i·13^-s + 0.621i·17^-s + (−0.602 + 0.798i)19^-s − 0.556i·23^-s − 0.0852·25^-s − 1.65·29^-s − 0.320i·31^-s + 1.71i·35^-s + 0.507i·37^-s + 1.16·41^-s + 0.770·43^-s + 1.08i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$-0.303 - 0.952i$
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:	$(2,\ 2736,\ (\ :1/2),\ -0.303 - 0.952i)$

Particular Values

$L(1)$	$\approx$	$0.1721057853$
$L(\frac12)$	$\approx$	$0.1721057853$
$L(\frac{3}{2})$		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 + (2.62 - 3.47i)T $
good	5	$ 1 + 2.32iT - 5T^{2} $
	7	$ 1 + 4.34T + 7T^{2} $
	11	$ 1 + 0.264iT - 11T^{2} $
	13	$ 1 + 6.38iT - 13T^{2} $
	17	$ 1 - 2.56iT - 17T^{2} $
	23	$ 1 + 2.66iT - 23T^{2} $
	29	$ 1 + 8.92T + 29T^{2} $
	31	$ 1 + 1.78iT - 31T^{2} $
	37	$ 1 - 3.08iT - 37T^{2} $
	41	$ 1 - 7.44T + 41T^{2} $
	43	$ 1 - 5.05T + 43T^{2} $
	47	$ 1 - 7.45iT - 47T^{2} $
	53	$ 1 + 4.10T + 53T^{2} $
	59	$ 1 + 8.36T + 59T^{2} $
	61	$ 1 + 8.45T + 61T^{2} $
	67	$ 1 + 2.35iT - 67T^{2} $
	71	$ 1 - 5.02T + 71T^{2} $
	73	$ 1 - 14.9T + 73T^{2} $
	79	$ 1 - 6.23iT - 79T^{2} $
	83	$ 1 - 16.0iT - 83T^{2} $
	89	$ 1 + 8.25T + 89T^{2} $
	97	$ 1 - 15.1iT - 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

−9.331188999132419992818371535096, −8.161306730909240405268418689526, −7.81771882771463500446587475113, −6.58331182444326146507780890525, −5.92989141484071477222576354581, −5.36448450935699137971086767462, −4.19610418498363643480741742566, −3.46376255144391872389964437439, −2.55063481445263247448761028412, −1.04020839903903565628283428557, 0.06194571828136504066417995844, 2.01286316225656687623527404960, 2.89372062001861472549384992964, 3.65479114581979646971640782261, 4.48383042753840882267731134272, 5.76993722695873490291814951853, 6.47834996288279379013179150791, 7.00243887246510848735266166940, 7.46939146640270388188904368471, 8.987026312734536039010048775060

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.1721057853\)
\(L(\frac12)\)	\(\approx\)	\(0.1721057853\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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