LMFDB - L-function 2-2736-228.227-c2-0-29

Dirichlet series

L(s) = 1

+ 1.26i·5^-s + 8.06i·7^-s + 8.70·11^-s + 16.8i·13^-s + 4.74i·17^-s + (5.61 + 18.1i)19^-s + 16.7·23^-s + 23.4·25^-s + 51.4·29^-s − 27.6·31^-s − 10.1·35^-s − 24.8i·37^-s − 13.7·41^-s − 5.53i·43^-s + 47.0·47^-s + ⋯

L(s) = 1

+ 0.252i·5^-s + 1.15i·7^-s + 0.790·11^-s + 1.29i·13^-s + 0.279i·17^-s + (0.295 + 0.955i)19^-s + 0.730·23^-s + 0.936·25^-s + 1.77·29^-s − 0.892·31^-s − 0.291·35^-s − 0.671i·37^-s − 0.335·41^-s − 0.128i·43^-s + 1.00·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$-0.381 - 0.924i$
Analytic conductor:	$74.5506$
Root analytic conductor:	$8.63426$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (2735, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1),\ -0.381 - 0.924i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.199660709$
$L(\frac12)$	$\approx$	$2.199660709$
$L(2)$		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 $
	19	$ 1 + (-5.61 - 18.1i)T $
good	5	$ 1 - 1.26iT - 25T^{2} $
	7	$ 1 - 8.06iT - 49T^{2} $
	11	$ 1 - 8.70T + 121T^{2} $
	13	$ 1 - 16.8iT - 169T^{2} $
	17	$ 1 - 4.74iT - 289T^{2} $
	23	$ 1 - 16.7T + 529T^{2} $
	29	$ 1 - 51.4T + 841T^{2} $
	31	$ 1 + 27.6T + 961T^{2} $
	37	$ 1 + 24.8iT - 1.36e3T^{2} $
	41	$ 1 + 13.7T + 1.68e3T^{2} $
	43	$ 1 + 5.53iT - 1.84e3T^{2} $
	47	$ 1 - 47.0T + 2.20e3T^{2} $
	53	$ 1 + 105.T + 2.80e3T^{2} $
	59	$ 1 + 2.94iT - 3.48e3T^{2} $
	61	$ 1 + 1.19T + 3.72e3T^{2} $
	67	$ 1 - 123.T + 4.48e3T^{2} $
	71	$ 1 - 99.1iT - 5.04e3T^{2} $
	73	$ 1 + 79.3T + 5.32e3T^{2} $
	79	$ 1 + 122.T + 6.24e3T^{2} $
	83	$ 1 - 90.1T + 6.88e3T^{2} $
	89	$ 1 - 10.0T + 7.92e3T^{2} $
	97	$ 1 + 84.1iT - 9.40e3T^{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

−8.903084559390283208104315412219, −8.348964263886419810713569816425, −7.21841355633238608295197500491, −6.57823547891952660572535441816, −5.91112536763030669298235694630, −5.01350592824594679532136966514, −4.13182844692766750803919741047, −3.18126724314997948182958190938, −2.21522945502232565942127033408, −1.26818127942303506785091646764, 0.58955741588156742357406300792, 1.21496734304584669963940636323, 2.81569890763808671814084401483, 3.50677967793021108709043229421, 4.60530080787023970992678703348, 5.07111294165582378060684601939, 6.25611128478570837173251559797, 6.95740125774457304025469496180, 7.56694049632527490478951217453, 8.442488819554968511692961003363

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.199660709\)
\(L(\frac12)\)	\(\approx\)	\(2.199660709\)
\(L(2)\)		not available
\(L(1)\)		not available

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