LMFDB - L-function 2-1368-57.56-c1-0-7

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1368$ = $2^{3} \cdot 3^{2} \cdot 19$
Sign:	$0.303 - 0.952i$
Analytic conductor:	$10.9235$
Root analytic conductor:	$3.30507$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1368} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1368,\ (\ :1/2),\ 0.303 - 0.952i)$

Particular Values

$L(1)$	$\approx$	$1.963653959$
$L(\frac12)$	$\approx$	$1.963653959$
$L(\frac{3}{2})$		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

−9.680982495324993040052875690858, −9.011034544738821619556887307553, −7.964715543108766162788862539064, −7.37713927045492216632797902107, −6.60352942873779855573603152766, −5.58038782044077925962079747428, −4.61681421069477944984841477402, −3.80508355773161290146248071665, −2.45228934591399906798430130165, −1.57048878226440236046034666152, 0.874987583331841899485346854896, 1.88966024163281569896995683768, 3.34668080717882999149001198063, 4.55531474087652865672342287000, 5.21682490262067582341217144187, 5.73380836149251356438998001384, 7.29919094485546968102604814608, 7.944914979292557114380915282042, 8.499115195752807581347456647622, 9.270331993170375990440008645559

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

\(L(1)\)	\(\approx\)	\(1.963653959\)
\(L(\frac12)\)	\(\approx\)	\(1.963653959\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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