LMFDB - L-function 2-160-8.5-c7-0-15

Dirichlet series

L(s) = 1

+ 69.7i·3^-s − 125i·5^-s − 1.35e3·7^-s − 2.67e3·9^-s + 2.63e3i·11^-s + 1.51e4i·13^-s + 8.71e3·15^-s − 6.98e3·17^-s − 3.27e4i·19^-s − 9.47e4i·21^-s − 8.06e4·23^-s − 1.56e4·25^-s − 3.40e4i·27^-s − 9.94e4i·29^-s + 6.67e4·31^-s + ⋯

L(s) = 1

+ 1.49i·3^-s − 0.447i·5^-s − 1.49·7^-s − 1.22·9^-s + 0.597i·11^-s + 1.91i·13^-s + 0.666·15^-s − 0.345·17^-s − 1.09i·19^-s − 2.23i·21^-s − 1.38·23^-s − 0.199·25^-s − 0.333i·27^-s − 0.756i·29^-s + 0.402·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$160$ = $2^{5} \cdot 5$
Sign:	$0.598 + 0.801i$
Analytic conductor:	$49.9816$
Root analytic conductor:	$7.06976$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{160} (81, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 160,\ (\ :7/2),\ 0.598 + 0.801i)$

Particular Values

$L(4)$	$\approx$	$0.3292745245$
$L(\frac12)$	$\approx$	$0.3292745245$
$L(\frac{9}{2})$		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 + 125iT $
good	3	$ 1 - 69.7iT - 2.18e3T^{2} $
	7	$ 1 + 1.35e3T + 8.23e5T^{2} $
	11	$ 1 - 2.63e3iT - 1.94e7T^{2} $
	13	$ 1 - 1.51e4iT - 6.27e7T^{2} $
	17	$ 1 + 6.98e3T + 4.10e8T^{2} $
	19	$ 1 + 3.27e4iT - 8.93e8T^{2} $
	23	$ 1 + 8.06e4T + 3.40e9T^{2} $
	29	$ 1 + 9.94e4iT - 1.72e10T^{2} $
	31	$ 1 - 6.67e4T + 2.75e10T^{2} $
	37	$ 1 + 2.87e5iT - 9.49e10T^{2} $
	41	$ 1 - 7.76e5T + 1.94e11T^{2} $
	43	$ 1 + 4.46e5iT - 2.71e11T^{2} $
	47	$ 1 - 3.79e5T + 5.06e11T^{2} $
	53	$ 1 + 8.12e4iT - 1.17e12T^{2} $
	59	$ 1 + 6.33e5iT - 2.48e12T^{2} $
	61	$ 1 - 9.78e5iT - 3.14e12T^{2} $
	67	$ 1 - 2.57e6iT - 6.06e12T^{2} $
	71	$ 1 - 3.86e6T + 9.09e12T^{2} $
	73	$ 1 + 5.37e6T + 1.10e13T^{2} $
	79	$ 1 + 8.07e5T + 1.92e13T^{2} $
	83	$ 1 + 2.70e6iT - 2.71e13T^{2} $
	89	$ 1 - 3.28e6T + 4.42e13T^{2} $
	97	$ 1 - 4.89e6T + 8.07e13T^{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

−11.31335801595519455040496839112, −10.10456482227555541044229404745, −9.435771157973831327617892034603, −8.930841446978051799952598049784, −7.05219611424322215434831913433, −5.94100857687896742666103572551, −4.45019791016368003992583136470, −3.95023437954654880705806220662, −2.38057782306533037341834772136, −0.10436219221722409579026139521, 0.942358875913096287265850195779, 2.56567762184557727726585780444, 3.45826732564418757851348316009, 5.88606104334878942329042842149, 6.31710246284033487019632820608, 7.53236033225842245768316407133, 8.264879668754269157493481148792, 9.800263722267088095889926043136, 10.69998930381848233174278071307, 12.11735277326556744507106035727

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 $\F_{q}$
Belyi maps

\(L(4)\)	\(\approx\)	\(0.3292745245\)
\(L(\frac12)\)	\(\approx\)	\(0.3292745245\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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