LMFDB - L-function 2-5616-12.11-c1-0-26

Dirichlet series

L(s) = 1

− 2.52i·5^-s − 0.792i·7^-s + 0.627·11^-s + 13^-s + 3.46i·17^-s − 3.37·23^-s − 1.37·25^-s + 5.84i·29^-s + 7.42i·31^-s − 2·35^-s + 0.627·37^-s + 3.16i·41^-s + 9.94i·43^-s + 3.62·47^-s + 6.37·49^-s + ⋯

L(s) = 1

− 1.12i·5^-s − 0.299i·7^-s + 0.189·11^-s + 0.277·13^-s + 0.840i·17^-s − 0.703·23^-s − 0.274·25^-s + 1.08i·29^-s + 1.33i·31^-s − 0.338·35^-s + 0.103·37^-s + 0.494i·41^-s + 1.51i·43^-s + 0.529·47^-s + 0.910·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5616$ = $2^{4} \cdot 3^{3} \cdot 13$
Sign:	$0.866 - 0.5i$
Analytic conductor:	$44.8439$
Root analytic conductor:	$6.69656$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5616} (2159, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5616,\ (\ :1/2),\ 0.866 - 0.5i)$

Particular Values

$L(1)$	$\approx$	$1.662271696$
$L(\frac12)$	$\approx$	$1.662271696$
$L(\frac{3}{2})$		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 - T $
good	5	$ 1 + 2.52iT - 5T^{2} $
	7	$ 1 + 0.792iT - 7T^{2} $
	11	$ 1 - 0.627T + 11T^{2} $
	17	$ 1 - 3.46iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 3.37T + 23T^{2} $
	29	$ 1 - 5.84iT - 29T^{2} $
	31	$ 1 - 7.42iT - 31T^{2} $
	37	$ 1 - 0.627T + 37T^{2} $
	41	$ 1 - 3.16iT - 41T^{2} $
	43	$ 1 - 9.94iT - 43T^{2} $
	47	$ 1 - 3.62T + 47T^{2} $
	53	$ 1 - 11.1iT - 53T^{2} $
	59	$ 1 + 11.7T + 59T^{2} $
	61	$ 1 - 4.37T + 61T^{2} $
	67	$ 1 + 8.21iT - 67T^{2} $
	71	$ 1 - 9.11T + 71T^{2} $
	73	$ 1 - 1.25T + 73T^{2} $
	79	$ 1 - 0.294iT - 79T^{2} $
	83	$ 1 + 6.25T + 83T^{2} $
	89	$ 1 + 3.61iT - 89T^{2} $
	97	$ 1 - 11.4T + 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

−8.203807034482836623307528736493, −7.66881059968378386939275164666, −6.69384874594328393230875423699, −6.05465380112278079651842917596, −5.22352952914917925289381157662, −4.56350168467162047845850867105, −3.88123876125947419614500546659, −2.96702266774611522014869127205, −1.66034514458297819440918705373, −1.01921324672618546754819195503, 0.49158377246115051641895484165, 2.06933328491322996154053236950, 2.63907270920047957176595447048, 3.60247191213630308114947451333, 4.24297966958479947874133182985, 5.35697926660728290405198127057, 5.99040595513250416279541406171, 6.69373729670559220946745291701, 7.31202264806812000476896748166, 7.969650473658393386205139198311

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

\(L(1)\)	\(\approx\)	\(1.662271696\)
\(L(\frac12)\)	\(\approx\)	\(1.662271696\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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