LMFDB - L-function 2-2268-21.20-c3-0-10

Dirichlet series

L(s) = 1

− 21.1·5^-s + (−13.5 − 12.5i)7^-s + 24.8i·11^-s + 60.6i·13^-s − 117.·17^-s + 104. i·19^-s − 20.1i·23^-s + 320.·25^-s + 28.0i·29^-s + 250. i·31^-s + (286. + 265. i)35^-s + 18.2·37^-s + 306.·41^-s − 149.·43^-s + 217.·47^-s + ⋯

L(s) = 1

− 1.88·5^-s + (−0.733 − 0.679i)7^-s + 0.681i·11^-s + 1.29i·13^-s − 1.67·17^-s + 1.26i·19^-s − 0.183i·23^-s + 2.56·25^-s + 0.179i·29^-s + 1.44i·31^-s + (1.38 + 1.28i)35^-s + 0.0809·37^-s + 1.16·41^-s − 0.528·43^-s + 0.674·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2268$ = $2^{2} \cdot 3^{4} \cdot 7$
Sign:	$-0.679 + 0.733i$
Analytic conductor:	$133.816$
Root analytic conductor:	$11.5679$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2268} (1133, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2268,\ (\ :3/2),\ -0.679 + 0.733i)$

Particular Values

$L(2)$	$\approx$	$0.3740642589$
$L(\frac12)$	$\approx$	$0.3740642589$
$L(\frac{5}{2})$		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 $
	7	$ 1 + (13.5 + 12.5i)T $
good	5	$ 1 + 21.1T + 125T^{2} $
	11	$ 1 - 24.8iT - 1.33e3T^{2} $
	13	$ 1 - 60.6iT - 2.19e3T^{2} $
	17	$ 1 + 117.T + 4.91e3T^{2} $
	19	$ 1 - 104. iT - 6.85e3T^{2} $
	23	$ 1 + 20.1iT - 1.21e4T^{2} $
	29	$ 1 - 28.0iT - 2.43e4T^{2} $
	31	$ 1 - 250. iT - 2.97e4T^{2} $
	37	$ 1 - 18.2T + 5.06e4T^{2} $
	41	$ 1 - 306.T + 6.89e4T^{2} $
	43	$ 1 + 149.T + 7.95e4T^{2} $
	47	$ 1 - 217.T + 1.03e5T^{2} $
	53	$ 1 + 116. iT - 1.48e5T^{2} $
	59	$ 1 + 76.7T + 2.05e5T^{2} $
	61	$ 1 - 570. iT - 2.26e5T^{2} $
	67	$ 1 - 67.6T + 3.00e5T^{2} $
	71	$ 1 - 796. iT - 3.57e5T^{2} $
	73	$ 1 - 710. iT - 3.89e5T^{2} $
	79	$ 1 - 80.1T + 4.93e5T^{2} $
	83	$ 1 + 115.T + 5.71e5T^{2} $
	89	$ 1 + 1.05e3T + 7.04e5T^{2} $
	97	$ 1 - 513. iT - 9.12e5T^{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

−8.954756565602882168717495663042, −8.428654387916850167562826154129, −7.41036425347189837641225888183, −7.02140706984385082987557220388, −6.37949743077628466488899251513, −4.85001495781838415930846396402, −4.08141474696177825578750197020, −3.83966852242896550757352148327, −2.59729294250080737036406398439, −1.19995794507054704561893343150, 0.16976374735942981162168226313, 0.48157038717277669065715265215, 2.54858577236242091483470694734, 3.18516883835447171354303872669, 4.05948496961823479421136176896, 4.82967652635998326761230053559, 5.89244271378790491471558947196, 6.71270080866761854046900586021, 7.51675644106227566308743409377, 8.192866848288752181254413070576

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.3740642589\)
\(L(\frac12)\)	\(\approx\)	\(0.3740642589\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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