LMFDB - L-function 2-252-3.2-c8-0-12

Dirichlet series

L(s) = 1

− 348. i·5^-s + 907.·7^-s + 1.18e4i·11^-s − 4.09e3·13^-s − 6.74e4i·17^-s − 1.16e5·19^-s + 6.83e4i·23^-s + 2.69e5·25^-s − 7.74e5i·29^-s + 8.07e5·31^-s − 3.16e5i·35^-s + 1.02e6·37^-s − 8.50e5i·41^-s − 3.69e6·43^-s + 5.47e6i·47^-s + ⋯

L(s) = 1

− 0.557i·5^-s + 0.377·7^-s + 0.812i·11^-s − 0.143·13^-s − 0.807i·17^-s − 0.896·19^-s + 0.244i·23^-s + 0.688·25^-s − 1.09i·29^-s + 0.874·31^-s − 0.210i·35^-s + 0.546·37^-s − 0.301i·41^-s − 1.07·43^-s + 1.12i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$252$ = $2^{2} \cdot 3^{2} \cdot 7$
Sign:	$-0.577 + 0.816i$
Analytic conductor:	$102.659$
Root analytic conductor:	$10.1320$
Motivic weight:	$8$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{252} (197, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 252,\ (\ :4),\ -0.577 + 0.816i)$

Particular Values

$L(\frac{9}{2})$	$\approx$	$1.248303394$
$L(\frac12)$	$\approx$	$1.248303394$
$L(5)$		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 $
	7	$ 1 - 907.T $
good	5	$ 1 + 348. iT - 3.90e5T^{2} $
	11	$ 1 - 1.18e4iT - 2.14e8T^{2} $
	13	$ 1 + 4.09e3T + 8.15e8T^{2} $
	17	$ 1 + 6.74e4iT - 6.97e9T^{2} $
	19	$ 1 + 1.16e5T + 1.69e10T^{2} $
	23	$ 1 - 6.83e4iT - 7.83e10T^{2} $
	29	$ 1 + 7.74e5iT - 5.00e11T^{2} $
	31	$ 1 - 8.07e5T + 8.52e11T^{2} $
	37	$ 1 - 1.02e6T + 3.51e12T^{2} $
	41	$ 1 + 8.50e5iT - 7.98e12T^{2} $
	43	$ 1 + 3.69e6T + 1.16e13T^{2} $
	47	$ 1 - 5.47e6iT - 2.38e13T^{2} $
	53	$ 1 - 5.41e5iT - 6.22e13T^{2} $
	59	$ 1 + 8.95e6iT - 1.46e14T^{2} $
	61	$ 1 - 6.50e5T + 1.91e14T^{2} $
	67	$ 1 + 3.39e7T + 4.06e14T^{2} $
	71	$ 1 + 1.63e7iT - 6.45e14T^{2} $
	73	$ 1 - 1.67e7T + 8.06e14T^{2} $
	79	$ 1 + 1.38e7T + 1.51e15T^{2} $
	83	$ 1 + 1.98e6iT - 2.25e15T^{2} $
	89	$ 1 + 6.05e7iT - 3.93e15T^{2} $
	97	$ 1 + 3.52e7T + 7.83e15T^{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

−10.19272908130299453048323505184, −9.327314933080286102750983212373, −8.345237530126528606777939103849, −7.40601472688550700243667468611, −6.28667992460836316699889637117, −4.97842196066054894075095450454, −4.31125736506723930615192609217, −2.70764530435457799444224461160, −1.53725599471069034070629966466, −0.27753035086624517822821297409, 1.17263121574121461282975275848, 2.50119040997186430547496988115, 3.61102101183806487925204913287, 4.83648723284473093393286655338, 6.06796535432910509446706925463, 6.91182972416550155492429170453, 8.157800184014339387278374292247, 8.842271170246342252620132106253, 10.25504580608190183388815868061, 10.83047060483565479366164936559

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 $\F_{q}$

\(L(\frac{9}{2})\)	\(\approx\)	\(1.248303394\)
\(L(\frac12)\)	\(\approx\)	\(1.248303394\)
\(L(5)\)		not available
\(L(1)\)		not available

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