LMFDB - L-function 2-2736-12.11-c1-0-22

Dirichlet series

L(s) = 1

− 0.363i·5^-s − 4.38i·7^-s + 5.87·11^-s + 5.79·13^-s + 6.55i·17^-s + i·19^-s − 1.26·23^-s + 4.86·25^-s − 5.50i·29^-s + 1.79i·31^-s − 1.59·35^-s − 6.76·37^-s + 7.61i·41^-s + 3.92i·43^-s + 9.53·47^-s + ⋯

L(s) = 1

− 0.162i·5^-s − 1.65i·7^-s + 1.77·11^-s + 1.60·13^-s + 1.59i·17^-s + 0.229i·19^-s − 0.263·23^-s + 0.973·25^-s − 1.02i·29^-s + 0.321i·31^-s − 0.268·35^-s − 1.11·37^-s + 1.18i·41^-s + 0.598i·43^-s + 1.39·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.816 + 0.577i$
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:	$(2,\ 2736,\ (\ :1/2),\ 0.816 + 0.577i)$

Particular Values

$L(1)$	$\approx$	$2.302246383$
$L(\frac12)$	$\approx$	$2.302246383$
$L(\frac{3}{2})$		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 + 0.363iT - 5T^{2} $
	7	$ 1 + 4.38iT - 7T^{2} $
	11	$ 1 - 5.87T + 11T^{2} $
	13	$ 1 - 5.79T + 13T^{2} $
	17	$ 1 - 6.55iT - 17T^{2} $
	23	$ 1 + 1.26T + 23T^{2} $
	29	$ 1 + 5.50iT - 29T^{2} $
	31	$ 1 - 1.79iT - 31T^{2} $
	37	$ 1 + 6.76T + 37T^{2} $
	41	$ 1 - 7.61iT - 41T^{2} $
	43	$ 1 - 3.92iT - 43T^{2} $
	47	$ 1 - 9.53T + 47T^{2} $
	53	$ 1 - 4.13iT - 53T^{2} $
	59	$ 1 - 12.3T + 59T^{2} $
	61	$ 1 + 0.895T + 61T^{2} $
	67	$ 1 + 6.97iT - 67T^{2} $
	71	$ 1 + 2.53T + 71T^{2} $
	73	$ 1 + 8.38T + 73T^{2} $
	79	$ 1 + 16.6iT - 79T^{2} $
	83	$ 1 + 1.26T + 83T^{2} $
	89	$ 1 - 8.04iT - 89T^{2} $
	97	$ 1 - 6.81T + 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

−8.667522985386951494260357317904, −8.108361049868044204791840304342, −7.12787383809685287320126935190, −6.42911704224611233273861333775, −5.97619309534067449908122847313, −4.47852267758578701501389859512, −3.93675726964603229414029040858, −3.46195085889516208622513238781, −1.55102708950185427130789848334, −1.02795777773014362923491531583, 1.10175439944812617385728847983, 2.23198429022216757063696137202, 3.22368458317640571268401851556, 4.01707619845239093423238795762, 5.20987273976766318485341600855, 5.78205873110401038230465472688, 6.65459538116323651531374380774, 7.13712962427093324488464881263, 8.643987187235149209517013845447, 8.791838046620009010641589813168

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.302246383\)
\(L(\frac12)\)	\(\approx\)	\(2.302246383\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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