LMFDB - L-function 2-160-5.4-c3-0-14

Dirichlet series

L(s) = 1

− 4.76i·3^-s + 11.1·5^-s − 15.5i·7^-s + 4.30·9^-s − 53.2i·15^-s − 74.3·21^-s − 207. i·23^-s + 125.·25^-s − 149. i·27^-s − 306·29^-s − 174. i·35^-s + 460.·41^-s + 30.9i·43^-s + 48.1·45^-s + 643. i·47^-s + ⋯

L(s) = 1

− 0.916i·3^-s + 0.999·5^-s − 0.842i·7^-s + 0.159·9^-s − 0.916i·15^-s − 0.772·21^-s − 1.88i·23^-s + 1.00·25^-s − 1.06i·27^-s − 1.95·29^-s − 0.842i·35^-s + 1.75·41^-s + 0.109i·43^-s + 0.159·45^-s + 1.99i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$160$ = $2^{5} \cdot 5$
Sign:	$i$
Analytic conductor:	$9.44030$
Root analytic conductor:	$3.07250$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{160} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 160,\ (\ :3/2),\ i)$

Particular Values

$L(2)$	$\approx$	$1.38631 - 1.38631i$
$L(\frac12)$	$\approx$	$1.38631 - 1.38631i$
$L(\frac{5}{2})$		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 - 11.1T $
good	3	$ 1 + 4.76iT - 27T^{2} $
	7	$ 1 + 15.5iT - 343T^{2} $
	11	$ 1 + 1.33e3T^{2} $
	13	$ 1 - 2.19e3T^{2} $
	17	$ 1 - 4.91e3T^{2} $
	19	$ 1 + 6.85e3T^{2} $
	23	$ 1 + 207. iT - 1.21e4T^{2} $
	29	$ 1 + 306T + 2.43e4T^{2} $
	31	$ 1 + 2.97e4T^{2} $
	37	$ 1 - 5.06e4T^{2} $
	41	$ 1 - 460.T + 6.89e4T^{2} $
	43	$ 1 - 30.9iT - 7.95e4T^{2} $
	47	$ 1 - 643. iT - 1.03e5T^{2} $
	53	$ 1 - 1.48e5T^{2} $
	59	$ 1 + 2.05e5T^{2} $
	61	$ 1 - 40.2T + 2.26e5T^{2} $
	67	$ 1 - 1.09e3iT - 3.00e5T^{2} $
	71	$ 1 + 3.57e5T^{2} $
	73	$ 1 - 3.89e5T^{2} $
	79	$ 1 + 4.93e5T^{2} $
	83	$ 1 - 1.14e3iT - 5.71e5T^{2} $
	89	$ 1 - 1.38e3T + 7.04e5T^{2} $
	97	$ 1 - 9.12e5T^{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

−12.61289108597088711463851000237, −11.08388398729350156450138184318, −10.19884772760449632941826778256, −9.142440889879378783652296202863, −7.76447838998563013933559703178, −6.86068672533443588908611754775, −5.90699923915792669497212227410, −4.32361141496976920208105231329, −2.36667406218455127757591975495, −0.997949833365966833715204881377, 1.93666941953532595376525967970, 3.59059949057182386486378014440, 5.13166426923616119496647256907, 5.89157886146161490282339699735, 7.42208168823084292373623505920, 9.084321433929143711189704304566, 9.488086514179841990883361147101, 10.50925174309259090151431033594, 11.52826403336226378199576198281, 12.78374071863924259370828546915

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.38631 - 1.38631i\)
\(L(\frac12)\)	\(\approx\)	\(1.38631 - 1.38631i\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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