LMFDB - L-function 2-3200-8.5-c1-0-67

Dirichlet series

L(s) = 1

− 1.41i·3^-s + 4.24·7^-s + 0.999·9^-s − 5.65i·11^-s + 2i·13^-s − 6·17^-s − 2.82i·19^-s − 6i·21^-s − 7.07·23^-s − 5.65i·27^-s − 4i·29^-s − 2.82·31^-s − 8.00·33^-s + 2i·37^-s + 2.82·39^-s + ⋯

L(s) = 1

− 0.816i·3^-s + 1.60·7^-s + 0.333·9^-s − 1.70i·11^-s + 0.554i·13^-s − 1.45·17^-s − 0.648i·19^-s − 1.30i·21^-s − 1.47·23^-s − 1.08i·27^-s − 0.742i·29^-s − 0.508·31^-s − 1.39·33^-s + 0.328i·37^-s + 0.452·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3200$ = $2^{7} \cdot 5^{2}$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$25.5521$
Root analytic conductor:	$5.05491$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3200} (1601, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3200,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.812931121$
$L(\frac12)$	$\approx$	$1.812931121$
$L(\frac{3}{2})$		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 + 1.41iT - 3T^{2} $
	7	$ 1 - 4.24T + 7T^{2} $
	11	$ 1 + 5.65iT - 11T^{2} $
	13	$ 1 - 2iT - 13T^{2} $
	17	$ 1 + 6T + 17T^{2} $
	19	$ 1 + 2.82iT - 19T^{2} $
	23	$ 1 + 7.07T + 23T^{2} $
	29	$ 1 + 4iT - 29T^{2} $
	31	$ 1 + 2.82T + 31T^{2} $
	37	$ 1 - 2iT - 37T^{2} $
	41	$ 1 + 8T + 41T^{2} $
	43	$ 1 + 1.41iT - 43T^{2} $
	47	$ 1 - 1.41T + 47T^{2} $
	53	$ 1 + 2iT - 53T^{2} $
	59	$ 1 + 2.82iT - 59T^{2} $
	61	$ 1 + 14iT - 61T^{2} $
	67	$ 1 - 4.24iT - 67T^{2} $
	71	$ 1 - 2.82T + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 - 16.9T + 79T^{2} $
	83	$ 1 - 12.7iT - 83T^{2} $
	89	$ 1 + 6T + 89T^{2} $
	97	$ 1 + 10T + 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

−8.253791596767381985888644082809, −7.83871015087830508650174741336, −6.81601646898341503195630100214, −6.31192599994629443816179421541, −5.34011654922641314625779430203, −4.52408103662973909781699198621, −3.74615023843914494026088102957, −2.29601995891341981858499192254, −1.72950783519311556131506506940, −0.52910810802708064070589152590, 1.63171887486124918380389279639, 2.16724578394476069021648893376, 3.73860134702209565605510752218, 4.48101294386227015340897161105, 4.81829835325462410953757372426, 5.67807430364811971678562893232, 6.87877925473185174344873265738, 7.51187324319687390111205967058, 8.204053792698981787811028967183, 8.976306483043863557754116686052

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.812931121\)
\(L(\frac12)\)	\(\approx\)	\(1.812931121\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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