LMFDB - L-function 2-2736-57.56-c1-0-37

Dirichlet series

L(s) = 1

− 2.60i·5^-s + 0.723·7^-s − 5.10i·11^-s − 5.91i·13^-s − 2.27i·17^-s + (4.22 + 1.08i)19^-s − 0.0606i·23^-s − 1.80·25^-s + 2.46·29^-s + 8.86i·31^-s − 1.88i·35^-s − 1.03i·37^-s − 10.9·41^-s − 0.413·43^-s − 1.94i·47^-s + ⋯

L(s) = 1

− 1.16i·5^-s + 0.273·7^-s − 1.53i·11^-s − 1.63i·13^-s − 0.552i·17^-s + (0.968 + 0.248i)19^-s − 0.0126i·23^-s − 0.361·25^-s + 0.458·29^-s + 1.59i·31^-s − 0.318i·35^-s − 0.170i·37^-s − 1.71·41^-s − 0.0630·43^-s − 0.284i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$-0.762 + 0.647i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ -0.762 + 0.647i)$

Particular Values

$L(1)$	$\approx$	$1.615726381$
$L(\frac12)$	$\approx$	$1.615726381$
$L(\frac{3}{2})$		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 + (-4.22 - 1.08i)T $
good	5	$ 1 + 2.60iT - 5T^{2} $
	7	$ 1 - 0.723T + 7T^{2} $
	11	$ 1 + 5.10iT - 11T^{2} $
	13	$ 1 + 5.91iT - 13T^{2} $
	17	$ 1 + 2.27iT - 17T^{2} $
	23	$ 1 + 0.0606iT - 23T^{2} $
	29	$ 1 - 2.46T + 29T^{2} $
	31	$ 1 - 8.86iT - 31T^{2} $
	37	$ 1 + 1.03iT - 37T^{2} $
	41	$ 1 + 10.9T + 41T^{2} $
	43	$ 1 + 0.413T + 43T^{2} $
	47	$ 1 + 1.94iT - 47T^{2} $
	53	$ 1 + 6.82T + 53T^{2} $
	59	$ 1 - 5.46T + 59T^{2} $
	61	$ 1 - 7.54T + 61T^{2} $
	67	$ 1 - 12.6iT - 67T^{2} $
	71	$ 1 - 12.3T + 71T^{2} $
	73	$ 1 + 14.3T + 73T^{2} $
	79	$ 1 + 9.40iT - 79T^{2} $
	83	$ 1 + 8.01iT - 83T^{2} $
	89	$ 1 - 16.2T + 89T^{2} $
	97	$ 1 + 0.783iT - 97T^{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

−8.402260389049814071972982938468, −8.123728619284209931748255837136, −7.09676974829296148760004501521, −6.01999000410348745979590538591, −5.22603859981872638297225401634, −4.96764870233878442416433694317, −3.52307928176310363685368043283, −2.97997358800452169046659380895, −1.33374204485688867728263089085, −0.55090954395474852785625634286, 1.67920290203810240071408746088, 2.38792198577594662740590181472, 3.52002490819544532605677760021, 4.38381100641427351848090001805, 5.11411720555277710927214649822, 6.39756119340949767982582236140, 6.79566016410413660685276467690, 7.45697067126483791264096921626, 8.232073591238478480840603441440, 9.377782532085381510255370677209

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.615726381\)
\(L(\frac12)\)	\(\approx\)	\(1.615726381\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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