LMFDB - L-function 2-1300-65.64-c1-0-19

Dirichlet series

L(s) = 1

− 2.26i·3^-s − 1.11·7^-s − 2.11·9^-s − 5.37i·11^-s + (1.26 − 3.37i)13^-s + 7.90i·19^-s + 2.52i·21^-s − 6.49i·23^-s − 2i·27^-s − 3.63·29^-s + 3.14i·31^-s − 12.1·33^-s − 7.40·37^-s + (−7.63 − 2.85i)39^-s + 4.75i·41^-s + ⋯

L(s) = 1

− 1.30i·3^-s − 0.421·7^-s − 0.705·9^-s − 1.62i·11^-s + (0.349 − 0.936i)13^-s + 1.81i·19^-s + 0.550i·21^-s − 1.35i·23^-s − 0.384i·27^-s − 0.675·29^-s + 0.565i·31^-s − 2.11·33^-s − 1.21·37^-s + (−1.22 − 0.456i)39^-s + 0.742i·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1300$ = $2^{2} \cdot 5^{2} \cdot 13$
Sign:	$-0.994 + 0.105i$
Analytic conductor:	$10.3805$
Root analytic conductor:	$3.22188$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1300} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1300,\ (\ :1/2),\ -0.994 + 0.105i)$

Particular Values

$L(1)$	$\approx$	$1.144170922$
$L(\frac12)$	$\approx$	$1.144170922$
$L(\frac{3}{2})$		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 + (-1.26 + 3.37i)T $
good	3	$ 1 + 2.26iT - 3T^{2} $
	7	$ 1 + 1.11T + 7T^{2} $
	11	$ 1 + 5.37iT - 11T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 - 7.90iT - 19T^{2} $
	23	$ 1 + 6.49iT - 23T^{2} $
	29	$ 1 + 3.63T + 29T^{2} $
	31	$ 1 - 3.14iT - 31T^{2} $
	37	$ 1 + 7.40T + 37T^{2} $
	41	$ 1 - 4.75iT - 41T^{2} $
	43	$ 1 + 4.78iT - 43T^{2} $
	47	$ 1 - 6.16T + 47T^{2} $
	53	$ 1 - 0.292iT - 53T^{2} $
	59	$ 1 + 11.3iT - 59T^{2} $
	61	$ 1 + 5.34T + 61T^{2} $
	67	$ 1 + 11.8T + 67T^{2} $
	71	$ 1 + 3.43iT - 71T^{2} $
	73	$ 1 - 4.59T + 73T^{2} $
	79	$ 1 - 10.8T + 79T^{2} $
	83	$ 1 + 7.40T + 83T^{2} $
	89	$ 1 - 14.8iT - 89T^{2} $
	97	$ 1 - 3.76T + 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

−8.999662598957129433292253284077, −8.191676132330399078956640761171, −7.82246148019469434120786964036, −6.63846734079698453589132482879, −6.12828531950334094057949542178, −5.38542608859915520552710335800, −3.75020927033707863401052893757, −2.96222359943498307246660196235, −1.63202583425899656706702994114, −0.47209481027513463578094777977, 1.88563042287382903623469608333, 3.19889958733815877689541497328, 4.21844171348275325713469279904, 4.70943761193993475291737210290, 5.67383958333756034565978141062, 6.92569679147410576981838048034, 7.38953922051431564580721819906, 8.902353214182896062086944893182, 9.341922826720273073623374225392, 9.855558250328310626631954536717

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.144170922\)
\(L(\frac12)\)	\(\approx\)	\(1.144170922\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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