LMFDB - L-function 2-1040-13.12-c1-0-14

Dirichlet series

L(s) = 1

− 0.369·3^-s + i·5^-s − 0.956i·7^-s − 2.86·9^-s − 2.58i·11^-s + (−0.369 + 3.58i)13^-s − 0.369i·15^-s + 6.81·17^-s − 5.49i·19^-s + 0.353i·21^-s + 6.80·23^-s − 25^-s + 2.16·27^-s + 6.60·29^-s + 5.40i·31^-s + ⋯

L(s) = 1

− 0.213·3^-s + 0.447i·5^-s − 0.361i·7^-s − 0.954·9^-s − 0.779i·11^-s + (−0.102 + 0.994i)13^-s − 0.0954i·15^-s + 1.65·17^-s − 1.26i·19^-s + 0.0771i·21^-s + 1.41·23^-s − 0.200·25^-s + 0.417·27^-s + 1.22·29^-s + 0.970i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1040$ = $2^{4} \cdot 5 \cdot 13$
Sign:	$0.994 + 0.102i$
Analytic conductor:	$8.30444$
Root analytic conductor:	$2.88174$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1040} (961, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1040,\ (\ :1/2),\ 0.994 + 0.102i)$

Particular Values

$L(1)$	$\approx$	$1.427021724$
$L(\frac12)$	$\approx$	$1.427021724$
$L(\frac{3}{2})$		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 $
	5	$ 1 - iT $
	13	$ 1 + (0.369 - 3.58i)T $
good	3	$ 1 + 0.369T + 3T^{2} $
	7	$ 1 + 0.956iT - 7T^{2} $
	11	$ 1 + 2.58iT - 11T^{2} $
	17	$ 1 - 6.81T + 17T^{2} $
	19	$ 1 + 5.49iT - 19T^{2} $
	23	$ 1 - 6.80T + 23T^{2} $
	29	$ 1 - 6.60T + 29T^{2} $
	31	$ 1 - 5.40iT - 31T^{2} $
	37	$ 1 + 3.69iT - 37T^{2} $
	41	$ 1 - 8.08iT - 41T^{2} $
	43	$ 1 - 0.723T + 43T^{2} $
	47	$ 1 + 12.6iT - 47T^{2} $
	53	$ 1 + 8.08T + 53T^{2} $
	59	$ 1 - 6.23iT - 59T^{2} $
	61	$ 1 - 14.2T + 61T^{2} $
	67	$ 1 - 7.50iT - 67T^{2} $
	71	$ 1 - 9.05iT - 71T^{2} $
	73	$ 1 + 8.12iT - 73T^{2} $
	79	$ 1 - 8.43T + 79T^{2} $
	83	$ 1 + 9.69iT - 83T^{2} $
	89	$ 1 + 2.08iT - 89T^{2} $
	97	$ 1 + 10.4iT - 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

−10.00600171981094995228766073304, −8.982455621245417188929662604606, −8.392611448244763055381026242129, −7.23162933988312158403592970585, −6.62253704992948863752779566004, −5.61400148641137494944140832206, −4.80005604825280256651053654060, −3.44611842308301117260050663485, −2.71666935818814592061093421325, −0.909615278231222492553000025074, 1.01477369927509755702581319044, 2.60361619045439909315687491659, 3.58514493224439699989057055107, 4.99521476187679994322074994306, 5.51948591936217651889162598450, 6.38437117893507088771942682550, 7.71647301124708943788250896035, 8.133415960358462770226380519386, 9.160594008433961363732023635797, 9.935974925202553451916289886562

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.427021724\)
\(L(\frac12)\)	\(\approx\)	\(1.427021724\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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