LMFDB - L-function 2-7488-12.11-c1-0-10

Dirichlet series

L(s) = 1

+ 0.852i·5^-s − 2.60i·7^-s − 0.524·11^-s + 13^-s + 5.70i·17^-s + 6.04i·19^-s − 2.63·23^-s + 4.27·25^-s + 4.24i·29^-s − 7.07i·31^-s + 2.22·35^-s − 6.86·37^-s + 3.68i·41^-s − 4.17i·43^-s + 5.80·47^-s + ⋯

L(s) = 1

+ 0.381i·5^-s − 0.985i·7^-s − 0.158·11^-s + 0.277·13^-s + 1.38i·17^-s + 1.38i·19^-s − 0.550·23^-s + 0.854·25^-s + 0.787i·29^-s − 1.27i·31^-s + 0.375·35^-s − 1.12·37^-s + 0.574i·41^-s − 0.637i·43^-s + 0.846·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7488$ = $2^{6} \cdot 3^{2} \cdot 13$
Sign:	$-0.577 - 0.816i$
Analytic conductor:	$59.7919$
Root analytic conductor:	$7.73252$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7488} (4031, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7488,\ (\ :1/2),\ -0.577 - 0.816i)$

Particular Values

$L(1)$	$\approx$	$0.9768596997$
$L(\frac12)$	$\approx$	$0.9768596997$
$L(\frac{3}{2})$		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 - 0.852iT - 5T^{2} $
	7	$ 1 + 2.60iT - 7T^{2} $
	11	$ 1 + 0.524T + 11T^{2} $
	17	$ 1 - 5.70iT - 17T^{2} $
	19	$ 1 - 6.04iT - 19T^{2} $
	23	$ 1 + 2.63T + 23T^{2} $
	29	$ 1 - 4.24iT - 29T^{2} $
	31	$ 1 + 7.07iT - 31T^{2} $
	37	$ 1 + 6.86T + 37T^{2} $
	41	$ 1 - 3.68iT - 41T^{2} $
	43	$ 1 + 4.17iT - 43T^{2} $
	47	$ 1 - 5.80T + 47T^{2} $
	53	$ 1 + 5.70iT - 53T^{2} $
	59	$ 1 + 3.16T + 59T^{2} $
	61	$ 1 + 6.79T + 61T^{2} $
	67	$ 1 + 1.57iT - 67T^{2} $
	71	$ 1 + 8.02T + 71T^{2} $
	73	$ 1 - 5.27T + 73T^{2} $
	79	$ 1 - 12.0iT - 79T^{2} $
	83	$ 1 + 14.9T + 83T^{2} $
	89	$ 1 - 6.50iT - 89T^{2} $
	97	$ 1 + 13.2T + 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

−8.188462477328441870594656317288, −7.37527974558766676826344820838, −6.82782301114575667691656174354, −6.03517236574798478091842918438, −5.49524528406716476090836946867, −4.34307627302321936502106894685, −3.86028862808964608410357785756, −3.17553464733569104880002832748, −1.99294389492836506046510099323, −1.18784177933225693582201966876, 0.23945685919540067240701351883, 1.42195598505986780827436905857, 2.59571175699940502736478583303, 2.96799632648673122733246809765, 4.19539295827996343626617414101, 4.96972029871081376035435968089, 5.39012517179180711661652104836, 6.24785180969696874111014383416, 6.98504802610373350093145292647, 7.59577641334496062670842383591

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9768596997\)
\(L(\frac12)\)	\(\approx\)	\(0.9768596997\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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