LMFDB - L-function 2-160-8.5-c3-0-10

Dirichlet series

L(s) = 1

− 7.99i·3^-s − 5i·5^-s − 9.93·7^-s − 36.9·9^-s − 60.0i·11^-s + 32.3i·13^-s − 39.9·15^-s + 12.5·17^-s + 116. i·19^-s + 79.4i·21^-s − 204.·23^-s − 25·25^-s + 79.7i·27^-s − 31.8i·29^-s + 81.0·31^-s + ⋯

L(s) = 1

− 1.53i·3^-s − 0.447i·5^-s − 0.536·7^-s − 1.36·9^-s − 1.64i·11^-s + 0.689i·13^-s − 0.688·15^-s + 0.178·17^-s + 1.41i·19^-s + 0.825i·21^-s − 1.85·23^-s − 0.200·25^-s + 0.568i·27^-s − 0.203i·29^-s + 0.469·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$160$ = $2^{5} \cdot 5$
Sign:	$-0.992 - 0.119i$
Analytic conductor:	$9.44030$
Root analytic conductor:	$3.07250$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{160} (81, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 160,\ (\ :3/2),\ -0.992 - 0.119i)$

Particular Values

$L(2)$	$\approx$	$0.0616549 + 1.02977i$
$L(\frac12)$	$\approx$	$0.0616549 + 1.02977i$
$L(\frac{5}{2})$		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 + 5iT $
good	3	$ 1 + 7.99iT - 27T^{2} $
	7	$ 1 + 9.93T + 343T^{2} $
	11	$ 1 + 60.0iT - 1.33e3T^{2} $
	13	$ 1 - 32.3iT - 2.19e3T^{2} $
	17	$ 1 - 12.5T + 4.91e3T^{2} $
	19	$ 1 - 116. iT - 6.85e3T^{2} $
	23	$ 1 + 204.T + 1.21e4T^{2} $
	29	$ 1 + 31.8iT - 2.43e4T^{2} $
	31	$ 1 - 81.0T + 2.97e4T^{2} $
	37	$ 1 + 270. iT - 5.06e4T^{2} $
	41	$ 1 - 330.T + 6.89e4T^{2} $
	43	$ 1 + 138. iT - 7.95e4T^{2} $
	47	$ 1 + 211.T + 1.03e5T^{2} $
	53	$ 1 + 437. iT - 1.48e5T^{2} $
	59	$ 1 + 181. iT - 2.05e5T^{2} $
	61	$ 1 + 472. iT - 2.26e5T^{2} $
	67	$ 1 - 71.8iT - 3.00e5T^{2} $
	71	$ 1 - 767.T + 3.57e5T^{2} $
	73	$ 1 - 261.T + 3.89e5T^{2} $
	79	$ 1 - 600.T + 4.93e5T^{2} $
	83	$ 1 + 612. iT - 5.71e5T^{2} $
	89	$ 1 - 503.T + 7.04e5T^{2} $
	97	$ 1 + 500.T + 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

−12.15228344858723104359020687665, −11.24626165682380129957650559379, −9.762096704110396084715520414366, −8.438258318689264250290766119901, −7.81291416275451301506837295952, −6.39944443079120526352261173703, −5.81791087773090000612385962247, −3.66808542838909220503787273797, −1.94454857875880512944723481110, −0.47168222771085088827445430975, 2.71975647914977109060262038235, 4.06372968842578384667876361029, 5.01734627506037273392304261305, 6.45341649364693277916527826576, 7.80708473594928678101027971249, 9.320438824229855619334033793642, 9.935301136816605012416301028973, 10.60219312932903634683144087389, 11.78187105085169741917813079270, 12.89013527340027596170327723415

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.0616549 + 1.02977i\)
\(L(\frac12)\)	\(\approx\)	\(0.0616549 + 1.02977i\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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