LMFDB - L-function 2-2600-65.64-c1-0-7

Dirichlet series

L(s) = 1

− 2.56i·3^-s − 4.56·7^-s − 3.56·9^-s + 3.12i·11^-s + (0.561 − 3.56i)13^-s − 0.561i·17^-s + 2i·19^-s + 11.6i·21^-s + 5.12i·23^-s + 1.43i·27^-s − 3.12·29^-s − 3.12i·31^-s + 8·33^-s − 5.43·37^-s + (−9.12 − 1.43i)39^-s + ⋯

L(s) = 1

− 1.47i·3^-s − 1.72·7^-s − 1.18·9^-s + 0.941i·11^-s + (0.155 − 0.987i)13^-s − 0.136i·17^-s + 0.458i·19^-s + 2.54i·21^-s + 1.06i·23^-s + 0.276i·27^-s − 0.579·29^-s − 0.560i·31^-s + 1.39·33^-s − 0.894·37^-s + (−1.46 − 0.230i)39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2600$ = $2^{3} \cdot 5^{2} \cdot 13$
Sign:	$0.953 - 0.302i$
Analytic conductor:	$20.7611$
Root analytic conductor:	$4.55643$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2600} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2600,\ (\ :1/2),\ 0.953 - 0.302i)$

Particular Values

$L(1)$	$\approx$	$0.8046573705$
$L(\frac12)$	$\approx$	$0.8046573705$
$L(\frac{3}{2})$		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 + (-0.561 + 3.56i)T $
good	3	$ 1 + 2.56iT - 3T^{2} $
	7	$ 1 + 4.56T + 7T^{2} $
	11	$ 1 - 3.12iT - 11T^{2} $
	17	$ 1 + 0.561iT - 17T^{2} $
	19	$ 1 - 2iT - 19T^{2} $
	23	$ 1 - 5.12iT - 23T^{2} $
	29	$ 1 + 3.12T + 29T^{2} $
	31	$ 1 + 3.12iT - 31T^{2} $
	37	$ 1 + 5.43T + 37T^{2} $
	41	$ 1 - 8iT - 41T^{2} $
	43	$ 1 - 5.43iT - 43T^{2} $
	47	$ 1 - 3.43T + 47T^{2} $
	53	$ 1 + 4.24iT - 53T^{2} $
	59	$ 1 - 10iT - 59T^{2} $
	61	$ 1 - 11.1T + 61T^{2} $
	67	$ 1 - 0.876T + 67T^{2} $
	71	$ 1 - 5.68iT - 71T^{2} $
	73	$ 1 - 15.3T + 73T^{2} $
	79	$ 1 - 2.87T + 79T^{2} $
	83	$ 1 + 8.24T + 83T^{2} $
	89	$ 1 - 5.12iT - 89T^{2} $
	97	$ 1 - 10.2T + 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

−8.840943948702352432265089111461, −7.85923112273994063251208173439, −7.36217819014320460134796948639, −6.67798584467323267444568373829, −6.07173253856647655474903920706, −5.32974057763998084254725424915, −3.86497537265539858807967157663, −3.03393134292721245810844775875, −2.13903797390252975914510355764, −0.951815859639121313382064301598, 0.31841013166316998929969633249, 2.42195262904634865033279362704, 3.54894811178749053830617286293, 3.73160269945257836760988190966, 4.83266648377018802074424267658, 5.68472786711422659121344350904, 6.47446162178942725914020610640, 7.08167871366954240189159589361, 8.533618833437418159120353669660, 9.035874086904718129420060510833

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

\(L(1)\)	\(\approx\)	\(0.8046573705\)
\(L(\frac12)\)	\(\approx\)	\(0.8046573705\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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