LMFDB - L-function 2-288-24.11-c7-0-22

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$288$ = $2^{5} \cdot 3^{2}$
Sign:	$0.150 + 0.988i$
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:	$(2,\ 288,\ (\ :7/2),\ 0.150 + 0.988i)$

Particular Values

$L(4)$	$\approx$	$2.219684155$
$L(\frac12)$	$\approx$	$2.219684155$
$L(\frac{9}{2})$		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

−10.10357162410701663968234409529, −9.657093458021449594564974360367, −8.585911716655119118032291148719, −7.29309142668685838375951559038, −6.50357923701505705020345487768, −5.33291913014919751465238411650, −4.39898170270023322637634096195, −2.83293910991642597728522147487, −1.82594380042094756174189713128, −0.49184209594512069135699673710, 1.20379576623067824710513357282, 2.24440134554058535116060591440, 3.47900669933628892767956429019, 4.91945869890642460744826626469, 5.98479355286169732577104441620, 6.57234198201863246061686560931, 8.205729236355657797062621798090, 8.790806952636330682298386310333, 10.03447917184722517553833576465, 10.54868622868160474710073169260

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(2.219684155\)
\(L(\frac12)\)	\(\approx\)	\(2.219684155\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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