LMFDB - L-function 2-30e2-5.4-c5-0-7

Dirichlet series

L(s) = 1

− 88i·7^-s − 540·11^-s + 418i·13^-s − 594i·17^-s − 836·19^-s − 4.10e3i·23^-s − 594·29^-s + 4.25e3·31^-s − 298i·37^-s − 1.72e4·41^-s + 1.21e4i·43^-s + 1.29e3i·47^-s + 9.06e3·49^-s + 1.94e4i·53^-s − 7.66e3·59^-s + ⋯

L(s) = 1

− 0.678i·7^-s − 1.34·11^-s + 0.685i·13^-s − 0.498i·17^-s − 0.531·19^-s − 1.61i·23^-s − 0.131·29^-s + 0.795·31^-s − 0.0357i·37^-s − 1.60·41^-s + 0.997i·43^-s + 0.0855i·47^-s + 0.539·49^-s + 0.953i·53^-s − 0.286·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$900$ = $2^{2} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$144.345$
Root analytic conductor:	$12.0143$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{900} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 900,\ (\ :5/2),\ 0.447 - 0.894i)$

Particular Values

$L(3)$	$\approx$	$1.031853890$
$L(\frac12)$	$\approx$	$1.031853890$
$L(\frac{7}{2})$		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 $
	5	$ 1 $
good	7	$ 1 + 88iT - 1.68e4T^{2} $
	11	$ 1 + 540T + 1.61e5T^{2} $
	13	$ 1 - 418iT - 3.71e5T^{2} $
	17	$ 1 + 594iT - 1.41e6T^{2} $
	19	$ 1 + 836T + 2.47e6T^{2} $
	23	$ 1 + 4.10e3iT - 6.43e6T^{2} $
	29	$ 1 + 594T + 2.05e7T^{2} $
	31	$ 1 - 4.25e3T + 2.86e7T^{2} $
	37	$ 1 + 298iT - 6.93e7T^{2} $
	41	$ 1 + 1.72e4T + 1.15e8T^{2} $
	43	$ 1 - 1.21e4iT - 1.47e8T^{2} $
	47	$ 1 - 1.29e3iT - 2.29e8T^{2} $
	53	$ 1 - 1.94e4iT - 4.18e8T^{2} $
	59	$ 1 + 7.66e3T + 7.14e8T^{2} $
	61	$ 1 + 3.47e4T + 8.44e8T^{2} $
	67	$ 1 - 2.18e4iT - 1.35e9T^{2} $
	71	$ 1 - 4.68e4T + 1.80e9T^{2} $
	73	$ 1 + 6.75e4iT - 2.07e9T^{2} $
	79	$ 1 - 7.69e4T + 3.07e9T^{2} $
	83	$ 1 - 6.77e4iT - 3.93e9T^{2} $
	89	$ 1 - 2.97e4T + 5.58e9T^{2} $
	97	$ 1 + 1.22e5iT - 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

−9.599947340749524307770283161625, −8.574881558706222273816302232123, −7.86055489867944440276876455033, −6.96836180139172492263137897976, −6.18769132308770942256721161814, −4.94972971909911513305244724992, −4.34674078828028467722374994307, −3.05855191116269891629839567893, −2.12426163975535744952458126415, −0.73163819836755203867357653958, 0.26587996075051842240415559431, 1.75698941245322813925903634857, 2.73957655313444446460638721807, 3.69213032833222609434718102039, 5.09103315033385085129811913772, 5.55772141231612720390430138622, 6.60395728824762022453159269160, 7.74288631113524596028535444964, 8.259185652735669874548909017127, 9.196774681840523351866965026866

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.031853890\)
\(L(\frac12)\)	\(\approx\)	\(1.031853890\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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