LMFDB - L-function 2-2736-19.18-c2-0-43

Dirichlet series

L(s) = 1

− 5.26·5^-s + 1.82·7^-s + 15.7·11^-s + 14.8i·13^-s + 17.7·17^-s + (−10.9 + 15.5i)19^-s + 44.0·23^-s + 2.72·25^-s − 0.665i·29^-s − 45.5i·31^-s − 9.58·35^-s − 7.42i·37^-s − 55.1i·41^-s − 57.7·43^-s − 59.0·47^-s + ⋯

L(s) = 1

− 1.05·5^-s + 0.260·7^-s + 1.42·11^-s + 1.14i·13^-s + 1.04·17^-s + (−0.574 + 0.818i)19^-s + 1.91·23^-s + 0.108·25^-s − 0.0229i·29^-s − 1.47i·31^-s − 0.273·35^-s − 0.200i·37^-s − 1.34i·41^-s − 1.34·43^-s − 1.25·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.574 - 0.818i$
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:	$(2,\ 2736,\ (\ :1),\ 0.574 - 0.818i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.827706272$
$L(\frac12)$	$\approx$	$1.827706272$
$L(2)$		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 + (10.9 - 15.5i)T $
good	5	$ 1 + 5.26T + 25T^{2} $
	7	$ 1 - 1.82T + 49T^{2} $
	11	$ 1 - 15.7T + 121T^{2} $
	13	$ 1 - 14.8iT - 169T^{2} $
	17	$ 1 - 17.7T + 289T^{2} $
	23	$ 1 - 44.0T + 529T^{2} $
	29	$ 1 + 0.665iT - 841T^{2} $
	31	$ 1 + 45.5iT - 961T^{2} $
	37	$ 1 + 7.42iT - 1.36e3T^{2} $
	41	$ 1 + 55.1iT - 1.68e3T^{2} $
	43	$ 1 + 57.7T + 1.84e3T^{2} $
	47	$ 1 + 59.0T + 2.20e3T^{2} $
	53	$ 1 - 45.8iT - 2.80e3T^{2} $
	59	$ 1 - 98.2iT - 3.48e3T^{2} $
	61	$ 1 - 90.9T + 3.72e3T^{2} $
	67	$ 1 + 8.94iT - 4.48e3T^{2} $
	71	$ 1 - 39.2iT - 5.04e3T^{2} $
	73	$ 1 - 42.6T + 5.32e3T^{2} $
	79	$ 1 - 91.1iT - 6.24e3T^{2} $
	83	$ 1 - 40.0T + 6.88e3T^{2} $
	89	$ 1 + 20.1iT - 7.92e3T^{2} $
	97	$ 1 + 120. iT - 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

−8.732147164316173641144131297088, −8.031353499905733060396829559361, −7.20676330305250158938396385862, −6.66053690083042275384024207377, −5.71503641619176529875932271622, −4.63715707331309046871702167060, −3.96057620156203338740038140357, −3.36930449154754037378956253799, −1.91420393054056952824151588381, −0.925668243894936368193744761475, 0.56330080915936538741634323282, 1.50463502280880051430971119284, 3.20687338790189254453385018429, 3.45579844632931321461441569807, 4.72297338695116121069151480612, 5.16482132828864906178197289803, 6.52428363805928965153954979191, 6.91315409376556714809040445377, 7.964455516183405016081428057598, 8.347399839480816691486824419876

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.827706272\)
\(L(\frac12)\)	\(\approx\)	\(1.827706272\)
\(L(2)\)		not available
\(L(1)\)		not available

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