LMFDB - L-function 2-832-13.12-c3-0-20

Dirichlet series

L(s) = 1

− 3.21·3^-s + 9.41i·5^-s − 19.8i·7^-s − 16.6·9^-s + 18.4i·11^-s + (29.1 − 36.7i)13^-s − 30.3i·15^-s − 128.·17^-s + 41.3i·19^-s + 63.9i·21^-s + 94.6·23^-s + 36.3·25^-s + 140.·27^-s − 61.9·29^-s − 167. i·31^-s + ⋯

L(s) = 1

− 0.619·3^-s + 0.842i·5^-s − 1.07i·7^-s − 0.616·9^-s + 0.505i·11^-s + (0.621 − 0.783i)13^-s − 0.521i·15^-s − 1.83·17^-s + 0.498i·19^-s + 0.664i·21^-s + 0.858·23^-s + 0.290·25^-s + 1.00·27^-s − 0.396·29^-s − 0.973i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$832$ = $2^{6} \cdot 13$
Sign:	$0.621 - 0.783i$
Analytic conductor:	$49.0895$
Root analytic conductor:	$7.00639$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{832} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 832,\ (\ :3/2),\ 0.621 - 0.783i)$

Particular Values

$L(2)$	$\approx$	$1.083610770$
$L(\frac12)$	$\approx$	$1.083610770$
$L(\frac{5}{2})$		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 + (-29.1 + 36.7i)T $
good	3	$ 1 + 3.21T + 27T^{2} $
	5	$ 1 - 9.41iT - 125T^{2} $
	7	$ 1 + 19.8iT - 343T^{2} $
	11	$ 1 - 18.4iT - 1.33e3T^{2} $
	17	$ 1 + 128.T + 4.91e3T^{2} $
	19	$ 1 - 41.3iT - 6.85e3T^{2} $
	23	$ 1 - 94.6T + 1.21e4T^{2} $
	29	$ 1 + 61.9T + 2.43e4T^{2} $
	31	$ 1 + 167. iT - 2.97e4T^{2} $
	37	$ 1 - 97.1iT - 5.06e4T^{2} $
	41	$ 1 + 390. iT - 6.89e4T^{2} $
	43	$ 1 + 185.T + 7.95e4T^{2} $
	47	$ 1 - 239. iT - 1.03e5T^{2} $
	53	$ 1 + 182.T + 1.48e5T^{2} $
	59	$ 1 - 745. iT - 2.05e5T^{2} $
	61	$ 1 - 356.T + 2.26e5T^{2} $
	67	$ 1 + 48.8iT - 3.00e5T^{2} $
	71	$ 1 - 735. iT - 3.57e5T^{2} $
	73	$ 1 + 388. iT - 3.89e5T^{2} $
	79	$ 1 - 1.12e3T + 4.93e5T^{2} $
	83	$ 1 - 657. iT - 5.71e5T^{2} $
	89	$ 1 - 827. iT - 7.04e5T^{2} $
	97	$ 1 - 1.70e3iT - 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

−10.28499131331035582482873062749, −9.139418607536025187203144842163, −8.169541909787268236911788482907, −7.12735501674429777078358075173, −6.59766297903202430709214053540, −5.65377169811499704167608220243, −4.54674260981229407102869358114, −3.55039588818905871515155340730, −2.40399401784031746511603457665, −0.77102460596221718688784929363, 0.44606573273294399292221768844, 1.88938089854278963872164713937, 3.12190259338151069365604768159, 4.62430178268621488746624982165, 5.20027234165420117933102476529, 6.19120110116501287424905022255, 6.80228612001744273281672667487, 8.502457030760723486653534566663, 8.722557526321902385102487116178, 9.400489933312112048457265585147

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

\(L(2)\)	\(\approx\)	\(1.083610770\)
\(L(\frac12)\)	\(\approx\)	\(1.083610770\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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