LMFDB - L-function 2-3200-5.4-c1-0-15

Dirichlet series

L(s) = 1

− 3.23i·3^-s + 1.23i·7^-s − 7.47·9^-s + 2·11^-s + 4.47i·13^-s − 4.47i·17^-s + 4.47·19^-s + 4.00·21^-s + 9.23i·23^-s + 14.4i·27^-s − 2·29^-s − 2.47·31^-s − 6.47i·33^-s + 10.9i·37^-s + 14.4·39^-s + ⋯

L(s) = 1

− 1.86i·3^-s + 0.467i·7^-s − 2.49·9^-s + 0.603·11^-s + 1.24i·13^-s − 1.08i·17^-s + 1.02·19^-s + 0.872·21^-s + 1.92i·23^-s + 2.78i·27^-s − 0.371·29^-s − 0.444·31^-s − 1.12i·33^-s + 1.79i·37^-s + 2.31·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3200$ = $2^{7} \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$25.5521$
Root analytic conductor:	$5.05491$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3200} (2049, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3200,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.616129106$
$L(\frac12)$	$\approx$	$1.616129106$
$L(\frac{3}{2})$		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 $
good	3	$ 1 + 3.23iT - 3T^{2} $
	7	$ 1 - 1.23iT - 7T^{2} $
	11	$ 1 - 2T + 11T^{2} $
	13	$ 1 - 4.47iT - 13T^{2} $
	17	$ 1 + 4.47iT - 17T^{2} $
	19	$ 1 - 4.47T + 19T^{2} $
	23	$ 1 - 9.23iT - 23T^{2} $
	29	$ 1 + 2T + 29T^{2} $
	31	$ 1 + 2.47T + 31T^{2} $
	37	$ 1 - 10.9iT - 37T^{2} $
	41	$ 1 - 3.52T + 41T^{2} $
	43	$ 1 - 5.70iT - 43T^{2} $
	47	$ 1 + 2.76iT - 47T^{2} $
	53	$ 1 + 8.47iT - 53T^{2} $
	59	$ 1 - 0.472T + 59T^{2} $
	61	$ 1 - 6T + 61T^{2} $
	67	$ 1 - 5.70iT - 67T^{2} $
	71	$ 1 - 6.47T + 71T^{2} $
	73	$ 1 - 4.47iT - 73T^{2} $
	79	$ 1 - 4.94T + 79T^{2} $
	83	$ 1 - 9.70iT - 83T^{2} $
	89	$ 1 + 2.94T + 89T^{2} $
	97	$ 1 + 7.52iT - 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

−8.463188882364411772512674431529, −7.67647664778189534177695510799, −7.07431124218652352012015004791, −6.61856467853935152677478566174, −5.72484890090561316661842639385, −5.08944600658640915641552319897, −3.64534787706388724842789829599, −2.71300350767611148305751110716, −1.78727367273853686502057781111, −1.05792465098334991198200364475, 0.58244596397165746558536183256, 2.46534055601039981576800215192, 3.49906492787997115279886284529, 3.95762231188293936270364078310, 4.74330701286207273816860408505, 5.58358872095124652103287658121, 6.11975700728254729465591274581, 7.35250151884258314139684702933, 8.213139665888124728216959958790, 8.914380939809229012006040088559

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

\(L(1)\)	\(\approx\)	\(1.616129106\)
\(L(\frac12)\)	\(\approx\)	\(1.616129106\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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