LMFDB - L-function 2-39e2-13.12-c1-0-21

Dirichlet series

L(s) = 1

− 2.69i·2^-s − 5.24·4^-s + 1.04i·5^-s − 0.554i·7^-s + 8.74i·8^-s + 2.82·10^-s + 2.91i·11^-s − 1.49·14^-s + 13.0·16^-s − 4.85·17^-s − 0.753i·19^-s − 5.50i·20^-s + 7.83·22^-s + 5.76·23^-s + 3.89·25^-s + ⋯

L(s) = 1

− 1.90i·2^-s − 2.62·4^-s + 0.469i·5^-s − 0.209i·7^-s + 3.09i·8^-s + 0.892·10^-s + 0.877i·11^-s − 0.399·14^-s + 3.25·16^-s − 1.17·17^-s − 0.172i·19^-s − 1.23i·20^-s + 1.67·22^-s + 1.20·23^-s + 0.779·25^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1521$ = $3^{2} \cdot 13^{2}$
Sign:	$-0.691 + 0.722i$
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:	$(2,\ 1521,\ (\ :1/2),\ -0.691 + 0.722i)$

Particular Values

$L(1)$	$\approx$	$1.275052131$
$L(\frac12)$	$\approx$	$1.275052131$
$L(\frac{3}{2})$		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 + 2.69iT - 2T^{2} $
	5	$ 1 - 1.04iT - 5T^{2} $
	7	$ 1 + 0.554iT - 7T^{2} $
	11	$ 1 - 2.91iT - 11T^{2} $
	17	$ 1 + 4.85T + 17T^{2} $
	19	$ 1 + 0.753iT - 19T^{2} $
	23	$ 1 - 5.76T + 23T^{2} $
	29	$ 1 - 1.91T + 29T^{2} $
	31	$ 1 + 9.51iT - 31T^{2} $
	37	$ 1 + 5.75iT - 37T^{2} $
	41	$ 1 + 4.91iT - 41T^{2} $
	43	$ 1 - 11.0T + 43T^{2} $
	47	$ 1 - 0.753iT - 47T^{2} $
	53	$ 1 - 7.58T + 53T^{2} $
	59	$ 1 - 4.09iT - 59T^{2} $
	61	$ 1 + 3.42T + 61T^{2} $
	67	$ 1 + 1.87iT - 67T^{2} $
	71	$ 1 - 10.5iT - 71T^{2} $
	73	$ 1 + 10.4iT - 73T^{2} $
	79	$ 1 - 1.33T + 79T^{2} $
	83	$ 1 + 2.64iT - 83T^{2} $
	89	$ 1 - 9.92iT - 89T^{2} $
	97	$ 1 - 17.0iT - 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

−9.144916118001328978364477725475, −9.068499512857250588079924617721, −7.72980211126192189781484376346, −6.84569275732665781395397430848, −5.50338455864197124873915890832, −4.48569166050070118406325560932, −3.96162832640416538757463410137, −2.72400930474258247374171246681, −2.17297939242587534594821052711, −0.74513726900819630907797377474, 0.889525709410778723954731107381, 3.10880848788406165685559886560, 4.36965224806786746449139638394, 5.00171811512600499612870916933, 5.78445809390531759087797132160, 6.60687633628678451520635001283, 7.17521574983361334149751064976, 8.215208383590223271517718146511, 8.826128470742642894608940990459, 9.084858814569638535857610145611

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.275052131\)
\(L(\frac12)\)	\(\approx\)	\(1.275052131\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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