LMFDB - L-function 2-39e2-13.12-c1-0-38

Dirichlet series

L(s) = 1

− 0.414i·2^-s + 1.82·4^-s + 2.82i·5^-s − 2.82i·7^-s − 1.58i·8^-s + 1.17·10^-s − 2i·11^-s − 1.17·14^-s + 3·16^-s + 7.65·17^-s − 2.82i·19^-s + 5.17i·20^-s − 0.828·22^-s − 4·23^-s − 3.00·25^-s + ⋯

L(s) = 1

− 0.292i·2^-s + 0.914·4^-s + 1.26i·5^-s − 1.06i·7^-s − 0.560i·8^-s + 0.370·10^-s − 0.603i·11^-s − 0.313·14^-s + 0.750·16^-s + 1.85·17^-s − 0.648i·19^-s + 1.15i·20^-s − 0.176·22^-s − 0.834·23^-s − 0.600·25^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1521$ = $3^{2} \cdot 13^{2}$
Sign:	$0.832 + 0.554i$
Analytic conductor:	$12.1452$
Root analytic conductor:	$3.48500$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1521} (1351, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1521,\ (\ :1/2),\ 0.832 + 0.554i)$

Particular Values

$L(1)$	$\approx$	$2.251771655$
$L(\frac12)$	$\approx$	$2.251771655$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	13	$ 1 $
good	2	$ 1 + 0.414iT - 2T^{2} $
	5	$ 1 - 2.82iT - 5T^{2} $
	7	$ 1 + 2.82iT - 7T^{2} $
	11	$ 1 + 2iT - 11T^{2} $
	17	$ 1 - 7.65T + 17T^{2} $
	19	$ 1 + 2.82iT - 19T^{2} $
	23	$ 1 + 4T + 23T^{2} $
	29	$ 1 + 2T + 29T^{2} $
	31	$ 1 + 1.17iT - 31T^{2} $
	37	$ 1 - 7.65iT - 37T^{2} $
	41	$ 1 + 5.17iT - 41T^{2} $
	43	$ 1 - 1.65T + 43T^{2} $
	47	$ 1 + 11.6iT - 47T^{2} $
	53	$ 1 - 2T + 53T^{2} $
	59	$ 1 - 7.65iT - 59T^{2} $
	61	$ 1 - 13.3T + 61T^{2} $
	67	$ 1 - 6.82iT - 67T^{2} $
	71	$ 1 + 2iT - 71T^{2} $
	73	$ 1 + 0.343iT - 73T^{2} $
	79	$ 1 + 11.3T + 79T^{2} $
	83	$ 1 + 3.65iT - 83T^{2} $
	89	$ 1 - 14.8iT - 89T^{2} $
	97	$ 1 - 3.65iT - 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

−9.951211457952290211831340763866, −8.421850496871375545892703314989, −7.45540704316005073855419970691, −7.12931186510186834148080617976, −6.28627702241700423378615421842, −5.46861574243499322804283991877, −3.85439872593714081772190906812, −3.32265561939173602520037418595, −2.39408715863243546139022398651, −0.997291238901048976908257670895, 1.32686348417768137272425893545, 2.27157440887970660545576082852, 3.48422170140855559161992741632, 4.75739521901296692030258979398, 5.67705588196388609971948333683, 5.94202440577143058534471705652, 7.29179986141208605329736553443, 7.965119972335520484408268960819, 8.583027754592137497839576006403, 9.544117159838564251674860213356

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.251771655\)
\(L(\frac12)\)	\(\approx\)	\(2.251771655\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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