LMFDB - L-function 2-3200-5.4-c1-0-62

Dirichlet series

L(s) = 1

− 0.414i·3^-s − 4.82i·7^-s + 2.82·9^-s + 3.24·11^-s − 5.65i·13^-s − 5.82i·17^-s + 4.41·19^-s − 1.99·21^-s − 0.828i·23^-s − 2.41i·27^-s − 8·29^-s + 4.82·31^-s − 1.34i·33^-s + 3.65i·37^-s − 2.34·39^-s + ⋯

L(s) = 1

− 0.239i·3^-s − 1.82i·7^-s + 0.942·9^-s + 0.977·11^-s − 1.56i·13^-s − 1.41i·17^-s + 1.01·19^-s − 0.436·21^-s − 0.172i·23^-s − 0.464i·27^-s − 1.48·29^-s + 0.867·31^-s − 0.233i·33^-s + 0.601i·37^-s − 0.375·39^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$3200$ = $2^{7} \cdot 5^{2}$
Sign:	$-0.447 + 0.894i$
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:	$(2,\ 3200,\ (\ :1/2),\ -0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$2.143350129$
$L(\frac12)$	$\approx$	$2.143350129$
$L(\frac{3}{2})$		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 + 0.414iT - 3T^{2} $
	7	$ 1 + 4.82iT - 7T^{2} $
	11	$ 1 - 3.24T + 11T^{2} $
	13	$ 1 + 5.65iT - 13T^{2} $
	17	$ 1 + 5.82iT - 17T^{2} $
	19	$ 1 - 4.41T + 19T^{2} $
	23	$ 1 + 0.828iT - 23T^{2} $
	29	$ 1 + 8T + 29T^{2} $
	31	$ 1 - 4.82T + 31T^{2} $
	37	$ 1 - 3.65iT - 37T^{2} $
	41	$ 1 + 0.656T + 41T^{2} $
	43	$ 1 - 10iT - 43T^{2} $
	47	$ 1 - 1.65iT - 47T^{2} $
	53	$ 1 - 3.65iT - 53T^{2} $
	59	$ 1 - 7.65T + 59T^{2} $
	61	$ 1 - 6T + 61T^{2} $
	67	$ 1 - 11.2iT - 67T^{2} $
	71	$ 1 - 1.65T + 71T^{2} $
	73	$ 1 + 0.171iT - 73T^{2} $
	79	$ 1 + 7.17T + 79T^{2} $
	83	$ 1 + 7.24iT - 83T^{2} $
	89	$ 1 + 11.8T + 89T^{2} $
	97	$ 1 - 5.31iT - 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

−8.164790691642322526428275558654, −7.45648829955846330235130076450, −7.17770097113962643612159828417, −6.37216528627293429639821053706, −5.26518934330761625167021583001, −4.45501572105016041421808354421, −3.73192579380079466697086674021, −2.90848804964381052085099897870, −1.27138145235014972874011789197, −0.75165534813702181885774385676, 1.61467278539112314076277788082, 2.12253965181989150401535365916, 3.55123709431033132021329775657, 4.13789733979090084155441852924, 5.15535957666994943401976620610, 5.88196006610654043788924276460, 6.60493868621492975119519607609, 7.30704806200788116225149302290, 8.430251612400801078090340495193, 8.967634256615479704314890621933

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.143350129\)
\(L(\frac12)\)	\(\approx\)	\(2.143350129\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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