LMFDB - L-function 2-2736-57.56-c1-0-35

Dirichlet series

L(s) = 1

− 3.42i·5^-s + 0.394·7^-s + 2.33i·11^-s − 3.36i·13^-s − 0.494i·17^-s + (0.300 − 4.34i)19^-s + 3.79i·23^-s − 6.75·25^-s − 2.14·29^-s − 7.04i·31^-s − 1.35i·35^-s − 7.30i·37^-s − 3.81·41^-s + 8.45·43^-s + 2.43i·47^-s + ⋯

L(s) = 1

− 1.53i·5^-s + 0.148·7^-s + 0.703i·11^-s − 0.933i·13^-s − 0.119i·17^-s + (0.0690 − 0.997i)19^-s + 0.790i·23^-s − 1.35·25^-s − 0.399·29^-s − 1.26i·31^-s − 0.228i·35^-s − 1.20i·37^-s − 0.595·41^-s + 1.28·43^-s + 0.355i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$-0.854 + 0.519i$
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:	$(2,\ 2736,\ (\ :1/2),\ -0.854 + 0.519i)$

Particular Values

$L(1)$	$\approx$	$1.252110573$
$L(\frac12)$	$\approx$	$1.252110573$
$L(\frac{3}{2})$		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 + (-0.300 + 4.34i)T $
good	5	$ 1 + 3.42iT - 5T^{2} $
	7	$ 1 - 0.394T + 7T^{2} $
	11	$ 1 - 2.33iT - 11T^{2} $
	13	$ 1 + 3.36iT - 13T^{2} $
	17	$ 1 + 0.494iT - 17T^{2} $
	23	$ 1 - 3.79iT - 23T^{2} $
	29	$ 1 + 2.14T + 29T^{2} $
	31	$ 1 + 7.04iT - 31T^{2} $
	37	$ 1 + 7.30iT - 37T^{2} $
	41	$ 1 + 3.81T + 41T^{2} $
	43	$ 1 - 8.45T + 43T^{2} $
	47	$ 1 - 2.43iT - 47T^{2} $
	53	$ 1 - 10.9T + 53T^{2} $
	59	$ 1 + 6.33T + 59T^{2} $
	61	$ 1 + 10.5T + 61T^{2} $
	67	$ 1 - 5.01iT - 67T^{2} $
	71	$ 1 + 0.759T + 71T^{2} $
	73	$ 1 - 3.06T + 73T^{2} $
	79	$ 1 + 6.40iT - 79T^{2} $
	83	$ 1 - 1.54iT - 83T^{2} $
	89	$ 1 + 13.2T + 89T^{2} $
	97	$ 1 - 1.71iT - 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

−8.578037119727711724939047844549, −7.75579290801117642672121847190, −7.25780692740202848030036880278, −5.95260372736816149223296277519, −5.33965956720419988986541678018, −4.64359370616140013202386486873, −3.90958825948284068890333752972, −2.61142149436816193658021478019, −1.49192649031112273210626903498, −0.40412873548519137010093337443, 1.56821298334394692046492779904, 2.67430494096353515221416694274, 3.41343477908871816314528786910, 4.24521116582898470742048243342, 5.39024975968739842988205767421, 6.34978038901521945739288409731, 6.69451043151546477325078045068, 7.56543576065118756783014880993, 8.318338555216506027405007508403, 9.118771579811087100180714020398

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.252110573\)
\(L(\frac12)\)	\(\approx\)	\(1.252110573\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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