LMFDB - L-function 2-432-12.11-c5-0-0

Dirichlet series

L(s) = 1

− 74.9i·5^-s − 119. i·7^-s − 389.·11^-s − 629·13^-s − 374. i·17^-s + 379. i·19^-s − 4.28e3·23^-s − 2.49e3·25^-s + 2.54e3i·29^-s + 4.74e3i·31^-s − 8.95e3·35^-s + 8.36e3·37^-s + 4.94e3i·41^-s − 9.48e3i·43^-s + 1.67e4·47^-s + ⋯

L(s) = 1

− 1.34i·5^-s − 0.921i·7^-s − 0.970·11^-s − 1.03·13^-s − 0.314i·17^-s + 0.241i·19^-s − 1.68·23^-s − 0.797·25^-s + 0.562i·29^-s + 0.887i·31^-s − 1.23·35^-s + 1.00·37^-s + 0.459i·41^-s − 0.782i·43^-s + 1.10·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$432$ = $2^{4} \cdot 3^{3}$
Sign:	$-i$
Analytic conductor:	$69.2858$
Root analytic conductor:	$8.32380$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{432} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 432,\ (\ :5/2),\ -i)$

Particular Values

$L(3)$	$\approx$	$0.1362961776$
$L(\frac12)$	$\approx$	$0.1362961776$
$L(\frac{7}{2})$		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 + 74.9iT - 3.12e3T^{2} $
	7	$ 1 + 119. iT - 1.68e4T^{2} $
	11	$ 1 + 389.T + 1.61e5T^{2} $
	13	$ 1 + 629T + 3.71e5T^{2} $
	17	$ 1 + 374. iT - 1.41e6T^{2} $
	19	$ 1 - 379. iT - 2.47e6T^{2} $
	23	$ 1 + 4.28e3T + 6.43e6T^{2} $
	29	$ 1 - 2.54e3iT - 2.05e7T^{2} $
	31	$ 1 - 4.74e3iT - 2.86e7T^{2} $
	37	$ 1 - 8.36e3T + 6.93e7T^{2} $
	41	$ 1 - 4.94e3iT - 1.15e8T^{2} $
	43	$ 1 + 9.48e3iT - 1.47e8T^{2} $
	47	$ 1 - 1.67e4T + 2.29e8T^{2} $
	53	$ 1 + 4.00e4iT - 4.18e8T^{2} $
	59	$ 1 - 1.59e4T + 7.14e8T^{2} $
	61	$ 1 - 1.67e4T + 8.44e8T^{2} $
	67	$ 1 - 6.02e4iT - 1.35e9T^{2} $
	71	$ 1 - 7.16e4T + 1.80e9T^{2} $
	73	$ 1 + 5.38e4T + 2.07e9T^{2} $
	79	$ 1 - 2.75e4iT - 3.07e9T^{2} $
	83	$ 1 + 6.15e4T + 3.93e9T^{2} $
	89	$ 1 + 1.14e5iT - 5.58e9T^{2} $
	97	$ 1 + 1.56e4T + 8.58e9T^{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

−10.31230648486743874165299219116, −9.828077348374339814759759570158, −8.657109018402408334329925366107, −7.891071862232344276438668978983, −7.04682221852144401876472788208, −5.57528806570239915904200647023, −4.82668429549984592128684350405, −3.90171550679689877275597595623, −2.31314617114203597449685238070, −0.953263820100425704452815425009, 0.03678291247804213795371778794, 2.29355445679787590242137405991, 2.71352937169203417804904498707, 4.17404625924925836700667624908, 5.54706796682710178208941105295, 6.27657795733835911686663968769, 7.43905146943158209651458448477, 8.069900941346458308019775032829, 9.414263110941832910411133343278, 10.16423241362059166727580612114

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(0.1362961776\)
\(L(\frac12)\)	\(\approx\)	\(0.1362961776\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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