LMFDB - L-function 2-2736-12.11-c1-0-25

Dirichlet series

L(s) = 1

− 1.28i·5^-s + 4.16i·7^-s − 0.743·11^-s − 4.42·13^-s − 1.23i·17^-s − i·19^-s − 7.13·23^-s + 3.35·25^-s − 7.86i·29^-s − 0.885i·31^-s + 5.34·35^-s + 0.287·37^-s − 7.60i·41^-s − 5.02i·43^-s + 9.88·47^-s + ⋯

L(s) = 1

− 0.573i·5^-s + 1.57i·7^-s − 0.224·11^-s − 1.22·13^-s − 0.298i·17^-s − 0.229i·19^-s − 1.48·23^-s + 0.671·25^-s − 1.46i·29^-s − 0.159i·31^-s + 0.903·35^-s + 0.0472·37^-s − 1.18i·41^-s − 0.766i·43^-s + 1.44·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$-0.0917 + 0.995i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (2015, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ -0.0917 + 0.995i)$

Particular Values

$L(1)$	$\approx$	$0.9275835602$
$L(\frac12)$	$\approx$	$0.9275835602$
$L(\frac{3}{2})$		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 + iT $
good	5	$ 1 + 1.28iT - 5T^{2} $
	7	$ 1 - 4.16iT - 7T^{2} $
	11	$ 1 + 0.743T + 11T^{2} $
	13	$ 1 + 4.42T + 13T^{2} $
	17	$ 1 + 1.23iT - 17T^{2} $
	23	$ 1 + 7.13T + 23T^{2} $
	29	$ 1 + 7.86iT - 29T^{2} $
	31	$ 1 + 0.885iT - 31T^{2} $
	37	$ 1 - 0.287T + 37T^{2} $
	41	$ 1 + 7.60iT - 41T^{2} $
	43	$ 1 + 5.02iT - 43T^{2} $
	47	$ 1 - 9.88T + 47T^{2} $
	53	$ 1 + 9.21iT - 53T^{2} $
	59	$ 1 - 10.9T + 59T^{2} $
	61	$ 1 + 11.2T + 61T^{2} $
	67	$ 1 + 7.62iT - 67T^{2} $
	71	$ 1 - 1.47T + 71T^{2} $
	73	$ 1 - 10.9T + 73T^{2} $
	79	$ 1 - 3.62iT - 79T^{2} $
	83	$ 1 - 7.21T + 83T^{2} $
	89	$ 1 - 13.4iT - 89T^{2} $
	97	$ 1 + 7.73T + 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

−8.609522904577471052497993396614, −8.040115899287371439442163878945, −7.16829663068585230044042254658, −6.15943980134628499905073115046, −5.44706269101456469630250282026, −4.90666851160655130160205443807, −3.88857154191187843675891572458, −2.52698341334433493035680669540, −2.13781728432284437440328061250, −0.31221418934481209932113557949, 1.16214469362784162019626799951, 2.46368274641043999901475660808, 3.43393199120998944477268966290, 4.26158874735824966055000631229, 4.99181402362600492746683437420, 6.11259031373020932746144103669, 6.90168468147687204032022429553, 7.47060903946912227148570534146, 8.004461017233051290964191305305, 9.100160641058018274173835228964

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9275835602\)
\(L(\frac12)\)	\(\approx\)	\(0.9275835602\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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