LMFDB - L-function 2-24e2-24.5-c6-0-24

Dirichlet series

L(s) = 1

− 168.·5^-s − 13.9·7^-s − 1.50e3·11^-s + 744. i·13^-s + 2.96e3i·17^-s + 6.31e3i·19^-s + 9.99e3i·23^-s + 1.29e4·25^-s − 2.03e4·29^-s + 3.92e4·31^-s + 2.36e3·35^-s + 8.92e4i·37^-s − 2.00e4i·41^-s − 5.54e4i·43^-s + 1.08e5i·47^-s + ⋯

L(s) = 1

− 1.35·5^-s − 0.0407·7^-s − 1.12·11^-s + 0.339i·13^-s + 0.603i·17^-s + 0.920i·19^-s + 0.821i·23^-s + 0.827·25^-s − 0.836·29^-s + 1.31·31^-s + 0.0551·35^-s + 1.76i·37^-s − 0.291i·41^-s − 0.697i·43^-s + 1.04i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(7-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$576$ = $2^{6} \cdot 3^{2}$
Sign:	$-0.169 + 0.985i$
Analytic conductor:	$132.511$
Root analytic conductor:	$11.5113$
Motivic weight:	$6$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{576} (161, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 576,\ (\ :3),\ -0.169 + 0.985i)$

Particular Values

$L(\frac{7}{2})$	$\approx$	$0.1095086610$
$L(\frac12)$	$\approx$	$0.1095086610$
$L(4)$		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 + 168.T + 1.56e4T^{2} $
	7	$ 1 + 13.9T + 1.17e5T^{2} $
	11	$ 1 + 1.50e3T + 1.77e6T^{2} $
	13	$ 1 - 744. iT - 4.82e6T^{2} $
	17	$ 1 - 2.96e3iT - 2.41e7T^{2} $
	19	$ 1 - 6.31e3iT - 4.70e7T^{2} $
	23	$ 1 - 9.99e3iT - 1.48e8T^{2} $
	29	$ 1 + 2.03e4T + 5.94e8T^{2} $
	31	$ 1 - 3.92e4T + 8.87e8T^{2} $
	37	$ 1 - 8.92e4iT - 2.56e9T^{2} $
	41	$ 1 + 2.00e4iT - 4.75e9T^{2} $
	43	$ 1 + 5.54e4iT - 6.32e9T^{2} $
	47	$ 1 - 1.08e5iT - 1.07e10T^{2} $
	53	$ 1 + 1.43e5T + 2.21e10T^{2} $
	59	$ 1 - 1.10e4T + 4.21e10T^{2} $
	61	$ 1 - 1.75e5iT - 5.15e10T^{2} $
	67	$ 1 - 4.20e4iT - 9.04e10T^{2} $
	71	$ 1 - 1.03e5iT - 1.28e11T^{2} $
	73	$ 1 - 5.39e5T + 1.51e11T^{2} $
	79	$ 1 + 4.79e5T + 2.43e11T^{2} $
	83	$ 1 + 4.85e5T + 3.26e11T^{2} $
	89	$ 1 - 3.10e5iT - 4.96e11T^{2} $
	97	$ 1 + 1.99e5T + 8.32e11T^{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

−9.596686733654750876256854913661, −8.230303909673649746888088783530, −7.992945909944226041486597144204, −6.99690811028973403563595344762, −5.84918306458885183468902316855, −4.74415015434591822096839158979, −3.83630516020657871823197508221, −2.92381203452800195369527916731, −1.46618464045408126280098384491, −0.03680253859188152651861359219, 0.61307676608589857421366187192, 2.42420339410647781707236171737, 3.34343909557699349403579858632, 4.45650501406221773745089049085, 5.22894206941643803494144120089, 6.56158296034701001818708714042, 7.56682265065579861984998841498, 8.041987454665006498233093591172, 9.014814863135783053198766448663, 10.10137667493089745927663809423

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(0.1095086610\)
\(L(\frac12)\)	\(\approx\)	\(0.1095086610\)
\(L(4)\)		not available
\(L(1)\)		not available

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