LMFDB - L-function 2-288-24.11-c7-0-12

Dirichlet series

L(s) = 1

− 294.·5^-s − 328. i·7^-s − 2.68e3i·11^-s + 1.48e3i·13^-s + 3.55e4i·17^-s − 1.60e4·19^-s − 1.15e4·23^-s + 8.46e3·25^-s − 2.04e5·29^-s − 1.49e5i·31^-s + 9.67e4i·35^-s − 6.15e4i·37^-s + 2.32e5i·41^-s − 6.19e5·43^-s + 1.07e6·47^-s + ⋯

L(s) = 1

− 1.05·5^-s − 0.362i·7^-s − 0.607i·11^-s + 0.187i·13^-s + 1.75i·17^-s − 0.538·19^-s − 0.198·23^-s + 0.108·25^-s − 1.55·29^-s − 0.901i·31^-s + 0.381i·35^-s − 0.199i·37^-s + 0.527i·41^-s − 1.18·43^-s + 1.51·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$288$ = $2^{5} \cdot 3^{2}$
Sign:	$0.982 + 0.187i$
Analytic conductor:	$89.9668$
Root analytic conductor:	$9.48508$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{288} (143, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 288,\ (\ :7/2),\ 0.982 + 0.187i)$

Particular Values

$L(4)$	$\approx$	$1.157952439$
$L(\frac12)$	$\approx$	$1.157952439$
$L(\frac{9}{2})$		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 $
	3	$ 1 $
good	5	$ 1 + 294.T + 7.81e4T^{2} $
	7	$ 1 + 328. iT - 8.23e5T^{2} $
	11	$ 1 + 2.68e3iT - 1.94e7T^{2} $
	13	$ 1 - 1.48e3iT - 6.27e7T^{2} $
	17	$ 1 - 3.55e4iT - 4.10e8T^{2} $
	19	$ 1 + 1.60e4T + 8.93e8T^{2} $
	23	$ 1 + 1.15e4T + 3.40e9T^{2} $
	29	$ 1 + 2.04e5T + 1.72e10T^{2} $
	31	$ 1 + 1.49e5iT - 2.75e10T^{2} $
	37	$ 1 + 6.15e4iT - 9.49e10T^{2} $
	41	$ 1 - 2.32e5iT - 1.94e11T^{2} $
	43	$ 1 + 6.19e5T + 2.71e11T^{2} $
	47	$ 1 - 1.07e6T + 5.06e11T^{2} $
	53	$ 1 + 7.13e5T + 1.17e12T^{2} $
	59	$ 1 + 1.30e6iT - 2.48e12T^{2} $
	61	$ 1 + 1.52e6iT - 3.14e12T^{2} $
	67	$ 1 - 4.81e6T + 6.06e12T^{2} $
	71	$ 1 - 4.36e6T + 9.09e12T^{2} $
	73	$ 1 + 1.23e6T + 1.10e13T^{2} $
	79	$ 1 - 5.34e6iT - 1.92e13T^{2} $
	83	$ 1 - 6.96e6iT - 2.71e13T^{2} $
	89	$ 1 - 1.66e5iT - 4.42e13T^{2} $
	97	$ 1 - 7.05e6T + 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

−10.81639595056018445620881537956, −9.622100825958798850054548031894, −8.389102381303525080319186187096, −7.86132553996087863926978717032, −6.69226659843863314148835583394, −5.63354621926090011042797150977, −4.11633319805803675208373496293, −3.63198995229020413716783730500, −1.95279127886797942017707615835, −0.49911396696268262911270129843, 0.53642170555848848673438967771, 2.16624582555715693397729180529, 3.41044240022667085000425240057, 4.49729206038938723442436576232, 5.51080328414507291650034912925, 6.97724507426556747844943961911, 7.61239459860518473287768124989, 8.703254576993402194290891236395, 9.588020260726426947085126806454, 10.72661732916410419404160888066

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(1.157952439\)
\(L(\frac12)\)	\(\approx\)	\(1.157952439\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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