LMFDB - L-function 2-2240-56.27-c1-0-37

Dirichlet series

L(s) = 1

− 0.237i·3^-s − 5^-s + (2.63 − 0.174i)7^-s + 2.94·9^-s + 4.69·11^-s + 2.77·13^-s + 0.237i·15^-s − 3.55i·17^-s + 6.72i·19^-s + (−0.0415 − 0.627i)21^-s − 8.45i·23^-s + 25^-s − 1.41i·27^-s + 5.80i·29^-s − 6.92·31^-s + ⋯

L(s) = 1

− 0.137i·3^-s − 0.447·5^-s + (0.997 − 0.0660i)7^-s + 0.981·9^-s + 1.41·11^-s + 0.770·13^-s + 0.0613i·15^-s − 0.862i·17^-s + 1.54i·19^-s + (−0.00906 − 0.136i)21^-s − 1.76i·23^-s + 0.200·25^-s − 0.271i·27^-s + 1.07i·29^-s − 1.24·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.980 + 0.194i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (671, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ 0.980 + 0.194i)$

Particular Values

$L(1)$	$\approx$	$2.311967806$
$L(\frac12)$	$\approx$	$2.311967806$
$L(\frac{3}{2})$		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 $
	5	$ 1 + T $
	7	$ 1 + (-2.63 + 0.174i)T $
good	3	$ 1 + 0.237iT - 3T^{2} $
	11	$ 1 - 4.69T + 11T^{2} $
	13	$ 1 - 2.77T + 13T^{2} $
	17	$ 1 + 3.55iT - 17T^{2} $
	19	$ 1 - 6.72iT - 19T^{2} $
	23	$ 1 + 8.45iT - 23T^{2} $
	29	$ 1 - 5.80iT - 29T^{2} $
	31	$ 1 + 6.92T + 31T^{2} $
	37	$ 1 - 0.171iT - 37T^{2} $
	41	$ 1 - 5.71iT - 41T^{2} $
	43	$ 1 + 3.26T + 43T^{2} $
	47	$ 1 - 2.83T + 47T^{2} $
	53	$ 1 - 7.09iT - 53T^{2} $
	59	$ 1 + 4.79iT - 59T^{2} $
	61	$ 1 + 9.63T + 61T^{2} $
	67	$ 1 - 9.99T + 67T^{2} $
	71	$ 1 + 7.49iT - 71T^{2} $
	73	$ 1 - 8.35iT - 73T^{2} $
	79	$ 1 + 5.05iT - 79T^{2} $
	83	$ 1 + 9.53iT - 83T^{2} $
	89	$ 1 + 5.75iT - 89T^{2} $
	97	$ 1 - 12.7iT - 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

−8.884374797280907610498209151982, −8.263162991288544528250762733604, −7.44969512647242605985079455405, −6.78481928244381198269798838196, −5.98147655264182337860728330217, −4.80687896161156309462297215736, −4.18774653288032234125093227680, −3.41894780485250979284364976556, −1.83510480135552373047154844503, −1.10015864214604735167733195213, 1.13745320324610001039784294719, 1.95025414325831113796300820156, 3.65516533875990690567551147617, 4.04221487252201846614596177833, 4.96186227568139644020600264309, 5.90997566719851080512697737562, 6.90234434995427301405896013144, 7.44365448029425478141217478882, 8.342504866396042804892240938708, 9.058829400064194271660739231912

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.311967806\)
\(L(\frac12)\)	\(\approx\)	\(2.311967806\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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