LMFDB - L-function 2-26e2-13.12-c3-0-18

Dirichlet series

L(s) = 1

+ 8.96·3^-s + 11.8i·5^-s + 32.1i·7^-s + 53.3·9^-s − 12.0i·11^-s + 105. i·15^-s − 89.7·17^-s + 19.1i·19^-s + 288. i·21^-s − 15.7·23^-s − 14.4·25^-s + 235.·27^-s + 262.·29^-s + 84.0i·31^-s − 108. i·33^-s + ⋯

L(s) = 1

+ 1.72·3^-s + 1.05i·5^-s + 1.73i·7^-s + 1.97·9^-s − 0.331i·11^-s + 1.82i·15^-s − 1.28·17^-s + 0.231i·19^-s + 2.99i·21^-s − 0.143·23^-s − 0.115·25^-s + 1.68·27^-s + 1.68·29^-s + 0.486i·31^-s − 0.571i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$676$ = $2^{2} \cdot 13^{2}$
Sign:	$-0.277 - 0.960i$
Analytic conductor:	$39.8852$
Root analytic conductor:	$6.31548$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{676} (337, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 676,\ (\ :3/2),\ -0.277 - 0.960i)$

Particular Values

$L(2)$	$\approx$	$3.546231535$
$L(\frac12)$	$\approx$	$3.546231535$
$L(\frac{5}{2})$		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 $
	13	$ 1 $
good	3	$ 1 - 8.96T + 27T^{2} $
	5	$ 1 - 11.8iT - 125T^{2} $
	7	$ 1 - 32.1iT - 343T^{2} $
	11	$ 1 + 12.0iT - 1.33e3T^{2} $
	17	$ 1 + 89.7T + 4.91e3T^{2} $
	19	$ 1 - 19.1iT - 6.85e3T^{2} $
	23	$ 1 + 15.7T + 1.21e4T^{2} $
	29	$ 1 - 262.T + 2.43e4T^{2} $
	31	$ 1 - 84.0iT - 2.97e4T^{2} $
	37	$ 1 + 271. iT - 5.06e4T^{2} $
	41	$ 1 - 165. iT - 6.89e4T^{2} $
	43	$ 1 + 270.T + 7.95e4T^{2} $
	47	$ 1 - 238. iT - 1.03e5T^{2} $
	53	$ 1 + 94.2T + 1.48e5T^{2} $
	59	$ 1 - 247. iT - 2.05e5T^{2} $
	61	$ 1 - 203.T + 2.26e5T^{2} $
	67	$ 1 + 231. iT - 3.00e5T^{2} $
	71	$ 1 + 749. iT - 3.57e5T^{2} $
	73	$ 1 + 153. iT - 3.89e5T^{2} $
	79	$ 1 + 881.T + 4.93e5T^{2} $
	83	$ 1 + 197. iT - 5.71e5T^{2} $
	89	$ 1 - 1.39e3iT - 7.04e5T^{2} $
	97	$ 1 - 1.12e3iT - 9.12e5T^{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

−10.15059510750065686058943298375, −9.202491235630204345823769017328, −8.688308481896100519150001880011, −8.020068729679298658000192550469, −6.90051238014420060671853235916, −6.11006943758165911046198731400, −4.66616385847146095179008266323, −3.30568106909150340190037958626, −2.69327063455450777375529771394, −1.97911526623975277120766299480, 0.76876582397568808405853295177, 1.88363695580056027963053773558, 3.20199947261063163300235696826, 4.28691714217134130948038051053, 4.67429183539786121385349164638, 6.71035881379015798780978610450, 7.35740369475095243366093742941, 8.381315865408144139722943266236, 8.700086540881154377179723792016, 9.824367539851042850866510968048

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(3.546231535\)
\(L(\frac12)\)	\(\approx\)	\(3.546231535\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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