LMFDB - L-function 2-3200-40.29-c1-0-55

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

$L(1)$	$\approx$	$0.6643328976$
$L(\frac12)$	$\approx$	$0.6643328976$
$L(\frac{3}{2})$		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

−8.308935960710054602172418091074, −7.33196952513675295407161116547, −6.68751795129918630418002375728, −5.98630894364181897922897366852, −5.36304032863979252518327431293, −4.80706392265289957854772789464, −3.74429926084785172068912350875, −2.66504344399960320604746914705, −0.964447248041121248330458733957, −0.35822190437345616787880044762, 1.27424355067160224852716213942, 2.09856075260930554006508240854, 3.94916117737574754190428489502, 4.41440218973449295562084928720, 5.25861125753129682366104801408, 5.97768648638132833399650763546, 6.58041521312045408377561461079, 7.30812792612669028754917165941, 7.976036381197546904797758051111, 9.202037167888371682219379941573

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6643328976\)
\(L(\frac12)\)	\(\approx\)	\(0.6643328976\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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