LMFDB - L-function 2-2736-19.18-c2-0-44

Dirichlet series

L(s) = 1

− 6.29·5^-s + 1.03·7^-s − 4.67·11^-s + 13.2i·13^-s − 9.09·17^-s + (−14.7 − 11.9i)19^-s − 16.2·23^-s + 14.5·25^-s + 12.4i·29^-s + 31.6i·31^-s − 6.52·35^-s − 57.9i·37^-s + 33.8i·41^-s + 31.3·43^-s + 14.7·47^-s + ⋯

L(s) = 1

− 1.25·5^-s + 0.148·7^-s − 0.424·11^-s + 1.01i·13^-s − 0.535·17^-s + (−0.777 − 0.628i)19^-s − 0.708·23^-s + 0.582·25^-s + 0.428i·29^-s + 1.02i·31^-s − 0.186·35^-s − 1.56i·37^-s + 0.826i·41^-s + 0.727·43^-s + 0.314·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.777 + 0.628i$
Analytic conductor:	$74.5506$
Root analytic conductor:	$8.63426$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (721, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1),\ 0.777 + 0.628i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.7889799444$
$L(\frac12)$	$\approx$	$0.7889799444$
$L(2)$		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 + (14.7 + 11.9i)T $
good	5	$ 1 + 6.29T + 25T^{2} $
	7	$ 1 - 1.03T + 49T^{2} $
	11	$ 1 + 4.67T + 121T^{2} $
	13	$ 1 - 13.2iT - 169T^{2} $
	17	$ 1 + 9.09T + 289T^{2} $
	23	$ 1 + 16.2T + 529T^{2} $
	29	$ 1 - 12.4iT - 841T^{2} $
	31	$ 1 - 31.6iT - 961T^{2} $
	37	$ 1 + 57.9iT - 1.36e3T^{2} $
	41	$ 1 - 33.8iT - 1.68e3T^{2} $
	43	$ 1 - 31.3T + 1.84e3T^{2} $
	47	$ 1 - 14.7T + 2.20e3T^{2} $
	53	$ 1 + 2.57iT - 2.80e3T^{2} $
	59	$ 1 + 43.3iT - 3.48e3T^{2} $
	61	$ 1 - 50.5T + 3.72e3T^{2} $
	67	$ 1 - 83.1iT - 4.48e3T^{2} $
	71	$ 1 - 8.54iT - 5.04e3T^{2} $
	73	$ 1 + 145.T + 5.32e3T^{2} $
	79	$ 1 - 100. iT - 6.24e3T^{2} $
	83	$ 1 - 139.T + 6.88e3T^{2} $
	89	$ 1 + 109. iT - 7.92e3T^{2} $
	97	$ 1 - 81.1iT - 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

−8.555671656673047430805258714353, −7.81558739710033412238833662972, −7.10120848482065964981510360756, −6.47247141596029079428885823711, −5.35451812228977021194571097856, −4.38927039995645880520537229173, −4.00110505369197559850055849891, −2.86323540347438792765503988336, −1.81298021298612099529001081269, −0.31903193265162440536094011131, 0.57862492764646963245810968792, 2.08642264123776498634600418204, 3.14864712098969387445185566317, 4.00248030598276170916722007404, 4.63888981729947231819120547366, 5.67329958679147613238193800024, 6.43286667856879596204982035835, 7.48913494178757282857424705067, 7.989165071876251317961678892783, 8.423155427600047485516237300876

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.7889799444\)
\(L(\frac12)\)	\(\approx\)	\(0.7889799444\)
\(L(2)\)		not available
\(L(1)\)		not available

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