LMFDB - L-function 2-3200-40.29-c1-0-57

Dirichlet series

L(s) = 1

+ 3.14·3^-s + 6.89·9^-s − 6.61i·11^-s + 7.89i·17^-s − 2.51i·19^-s + 12.2·27^-s − 20.7i·33^-s + 12.7·41^-s + 8.48·43^-s + 7·49^-s + 24.8i·51^-s − 7.89i·57^-s − 14.1i·59^-s − 7.88·67^-s − 13.6i·73^-s + ⋯

L(s) = 1

+ 1.81·3^-s + 2.29·9^-s − 1.99i·11^-s + 1.91i·17^-s − 0.575i·19^-s + 2.36·27^-s − 3.62i·33^-s + 1.99·41^-s + 1.29·43^-s + 49^-s + 3.48i·51^-s − 1.04i·57^-s − 1.84i·59^-s − 0.962·67^-s − 1.60i·73^-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:	$\chi_{3200} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3200,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$3.872015880$
$L(\frac12)$	$\approx$	$3.872015880$
$L(\frac{3}{2})$		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 - 3.14T + 3T^{2} $
	7	$ 1 - 7T^{2} $
	11	$ 1 + 6.61iT - 11T^{2} $
	13	$ 1 + 13T^{2} $
	17	$ 1 - 7.89iT - 17T^{2} $
	19	$ 1 + 2.51iT - 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 37T^{2} $
	41	$ 1 - 12.7T + 41T^{2} $
	43	$ 1 - 8.48T + 43T^{2} $
	47	$ 1 - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 + 14.1iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 7.88T + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 13.6iT - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 14.1T + 83T^{2} $
	89	$ 1 + 13.8T + 89T^{2} $
	97	$ 1 - 10iT - 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

−8.694456966113265450858356815617, −8.030979918585618768487476142105, −7.47509736126888225559663828783, −6.35748788928443585368827483395, −5.74043306232134414332336472567, −4.34128428266210633565032872472, −3.70056966729130069987874686345, −3.03850540824081876243881689517, −2.20462981083968075415354924776, −1.04915517856324537507270032164, 1.35739361210159566765396160787, 2.46096172149270170728816452702, 2.77953757995571206974801453813, 4.16819083496210740853678949335, 4.39991465280719747876510438221, 5.58681808862167487275126151904, 7.07835997738194773346562501417, 7.26655784388676931346745391755, 7.87484234375644257575952676319, 8.870942499135906625028042674216

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(3.872015880\)
\(L(\frac12)\)	\(\approx\)	\(3.872015880\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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