LMFDB - L-function 2-288-8.5-c7-0-3

Dirichlet series

L(s) = 1

− 425. i·5^-s − 1.66e3·7^-s + 2.46e3i·11^-s − 3.76e3i·13^-s − 1.62e4·17^-s − 4.86e3i·19^-s − 1.08e5·23^-s − 1.02e5·25^-s + 8.90e4i·29^-s + 6.94e4·31^-s + 7.07e5i·35^-s − 4.18e5i·37^-s − 2.74e5·41^-s + 4.62e5i·43^-s + 1.53e5·47^-s + ⋯

L(s) = 1

− 1.52i·5^-s − 1.83·7^-s + 0.558i·11^-s − 0.475i·13^-s − 0.801·17^-s − 0.162i·19^-s − 1.86·23^-s − 1.31·25^-s + 0.678i·29^-s + 0.418·31^-s + 2.79i·35^-s − 1.35i·37^-s − 0.621·41^-s + 0.887i·43^-s + 0.215·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$288$ = $2^{5} \cdot 3^{2}$
Sign:	$0.829 - 0.558i$
Analytic conductor:	$89.9668$
Root analytic conductor:	$9.48508$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{288} (145, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 288,\ (\ :7/2),\ 0.829 - 0.558i)$

Particular Values

$L(4)$	$\approx$	$0.5223380010$
$L(\frac12)$	$\approx$	$0.5223380010$
$L(\frac{9}{2})$		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 + 425. iT - 7.81e4T^{2} $
	7	$ 1 + 1.66e3T + 8.23e5T^{2} $
	11	$ 1 - 2.46e3iT - 1.94e7T^{2} $
	13	$ 1 + 3.76e3iT - 6.27e7T^{2} $
	17	$ 1 + 1.62e4T + 4.10e8T^{2} $
	19	$ 1 + 4.86e3iT - 8.93e8T^{2} $
	23	$ 1 + 1.08e5T + 3.40e9T^{2} $
	29	$ 1 - 8.90e4iT - 1.72e10T^{2} $
	31	$ 1 - 6.94e4T + 2.75e10T^{2} $
	37	$ 1 + 4.18e5iT - 9.49e10T^{2} $
	41	$ 1 + 2.74e5T + 1.94e11T^{2} $
	43	$ 1 - 4.62e5iT - 2.71e11T^{2} $
	47	$ 1 - 1.53e5T + 5.06e11T^{2} $
	53	$ 1 + 1.81e5iT - 1.17e12T^{2} $
	59	$ 1 - 6.47e5iT - 2.48e12T^{2} $
	61	$ 1 + 2.26e6iT - 3.14e12T^{2} $
	67	$ 1 + 2.31e6iT - 6.06e12T^{2} $
	71	$ 1 - 2.77e6T + 9.09e12T^{2} $
	73	$ 1 + 4.45e6T + 1.10e13T^{2} $
	79	$ 1 - 3.14e6T + 1.92e13T^{2} $
	83	$ 1 - 3.33e6iT - 2.71e13T^{2} $
	89	$ 1 - 5.00e6T + 4.42e13T^{2} $
	97	$ 1 - 8.06e6T + 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

−10.41415155252353737024771687000, −9.573144983427950876356124415998, −8.984352206301862356409780413900, −7.905787394130074511279923271888, −6.64006671757658033271087138152, −5.73839857375775704710031255976, −4.57565213077528529849079430972, −3.55641189709587319022402025488, −2.11204258346122917082341898164, −0.61610582270047099924719395989, 0.18484947167765182824508796086, 2.27862763233520047667091599675, 3.17237087230401423633176681643, 4.00547171925748620636741498260, 6.05296054723286454358364717902, 6.44043495127198970164518930727, 7.31416432545538799434702121862, 8.664321249842791906818914832651, 9.916749555619468727491715136867, 10.23404077780998558305216479843

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(0.5223380010\)
\(L(\frac12)\)	\(\approx\)	\(0.5223380010\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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