LMFDB - L-function 2-2600-13.12-c1-0-21

Dirichlet series

L(s) = 1

− 3.25·3^-s + 1.80i·7^-s + 7.57·9^-s − 0.556i·11^-s + (3.25 + 1.55i)13^-s + 2.77·17^-s + 7.83i·19^-s − 5.88i·21^-s − 0.140·23^-s − 14.8·27^-s + 1.92·29^-s − 6.21i·31^-s + 1.80i·33^-s − 10.3i·37^-s + (−10.5 − 5.06i)39^-s + ⋯

L(s) = 1

− 1.87·3^-s + 0.683i·7^-s + 2.52·9^-s − 0.167i·11^-s + (0.902 + 0.431i)13^-s + 0.671·17^-s + 1.79i·19^-s − 1.28i·21^-s − 0.0292·23^-s − 2.86·27^-s + 0.357·29^-s − 1.11i·31^-s + 0.314i·33^-s − 1.69i·37^-s + (−1.69 − 0.810i)39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2600$ = $2^{3} \cdot 5^{2} \cdot 13$
Sign:	$0.431 - 0.902i$
Analytic conductor:	$20.7611$
Root analytic conductor:	$4.55643$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2600} (2001, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2600,\ (\ :1/2),\ 0.431 - 0.902i)$

Particular Values

$L(1)$	$\approx$	$0.9244339294$
$L(\frac12)$	$\approx$	$0.9244339294$
$L(\frac{3}{2})$		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 $
	13	$ 1 + (-3.25 - 1.55i)T $
good	3	$ 1 + 3.25T + 3T^{2} $
	7	$ 1 - 1.80iT - 7T^{2} $
	11	$ 1 + 0.556iT - 11T^{2} $
	17	$ 1 - 2.77T + 17T^{2} $
	19	$ 1 - 7.83iT - 19T^{2} $
	23	$ 1 + 0.140T + 23T^{2} $
	29	$ 1 - 1.92T + 29T^{2} $
	31	$ 1 + 6.21iT - 31T^{2} $
	37	$ 1 + 10.3iT - 37T^{2} $
	41	$ 1 - 7.27iT - 41T^{2} $
	43	$ 1 - 9.13T + 43T^{2} $
	47	$ 1 + 7.34iT - 47T^{2} $
	53	$ 1 + 7.27T + 53T^{2} $
	59	$ 1 + 1.32iT - 59T^{2} $
	61	$ 1 + 5.97T + 61T^{2} $
	67	$ 1 - 8.46iT - 67T^{2} $
	71	$ 1 + 8.09iT - 71T^{2} $
	73	$ 1 + 4.92iT - 73T^{2} $
	79	$ 1 - 1.39T + 79T^{2} $
	83	$ 1 - 16.3iT - 83T^{2} $
	89	$ 1 - 0.382iT - 89T^{2} $
	97	$ 1 + 0.607iT - 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

−9.198390419862366097091403522697, −8.098603164096925464684324731801, −7.37991865569970915125471170787, −6.31061498703659246835732753036, −5.91546078000403565218692168030, −5.44101354267493145212745698432, −4.37389140774907849119811283121, −3.64586190411926140752794747279, −1.96085122348417804638863427533, −0.917330278798840938859130618492, 0.57692295739817187578174576463, 1.36381010084453848036887592501, 3.12272426733833935736116131050, 4.30533680498138751487400122699, 4.85806359636118636230933155463, 5.66844230980321908234861909482, 6.39336925953405484458365111681, 7.01508992778077467986169612145, 7.67122580439795105998831661771, 8.820584698734947864655879121958

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9244339294\)
\(L(\frac12)\)	\(\approx\)	\(0.9244339294\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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