LMFDB - L-function 2-2240-56.27-c1-0-44

Dirichlet series

L(s) = 1

− 2.59i·3^-s − 5^-s + (−2.28 + 1.33i)7^-s − 3.73·9^-s + 3.05·11^-s + 6.11·13^-s + 2.59i·15^-s − 1.25i·17^-s − 3.37i·19^-s + (3.47 + 5.92i)21^-s − 4.24i·23^-s + 25^-s + 1.92i·27^-s + 1.52i·29^-s + 9.46·31^-s + ⋯

L(s) = 1

− 1.49i·3^-s − 0.447·5^-s + (−0.862 + 0.505i)7^-s − 1.24·9^-s + 0.922·11^-s + 1.69·13^-s + 0.670i·15^-s − 0.304i·17^-s − 0.774i·19^-s + (0.757 + 1.29i)21^-s − 0.884i·23^-s + 0.200·25^-s + 0.369i·27^-s + 0.283i·29^-s + 1.70·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$-0.702 + 0.711i$
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:	$(2,\ 2240,\ (\ :1/2),\ -0.702 + 0.711i)$

Particular Values

$L(1)$	$\approx$	$1.445063772$
$L(\frac12)$	$\approx$	$1.445063772$
$L(\frac{3}{2})$		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 + (2.28 - 1.33i)T $
good	3	$ 1 + 2.59iT - 3T^{2} $
	11	$ 1 - 3.05T + 11T^{2} $
	13	$ 1 - 6.11T + 13T^{2} $
	17	$ 1 + 1.25iT - 17T^{2} $
	19	$ 1 + 3.37iT - 19T^{2} $
	23	$ 1 + 4.24iT - 23T^{2} $
	29	$ 1 - 1.52iT - 29T^{2} $
	31	$ 1 - 9.46T + 31T^{2} $
	37	$ 1 - 3.12iT - 37T^{2} $
	41	$ 1 + 3.19iT - 41T^{2} $
	43	$ 1 + 10.7T + 43T^{2} $
	47	$ 1 + 7.89T + 47T^{2} $
	53	$ 1 - 12.5iT - 53T^{2} $
	59	$ 1 + 14.5iT - 59T^{2} $
	61	$ 1 + 0.274T + 61T^{2} $
	67	$ 1 - 10.7T + 67T^{2} $
	71	$ 1 + 7.23iT - 71T^{2} $
	73	$ 1 - 0.742iT - 73T^{2} $
	79	$ 1 + 10.1iT - 79T^{2} $
	83	$ 1 + 6.15iT - 83T^{2} $
	89	$ 1 - 5.78iT - 89T^{2} $
	97	$ 1 + 14.3iT - 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

−8.542132436947477129570806718572, −8.088615730087958126330696325207, −6.92531618329069393991908356946, −6.50514176839229200066466757705, −6.09553546399615931063423494629, −4.76385239877835982856197624487, −3.57223296788467384961819685003, −2.79810482294062012539839607333, −1.58202267385946946149351541245, −0.58198204183354169626581421353, 1.23650793504342662556257420533, 3.16166333942104448950113295497, 3.81386456177716639254117100538, 4.09935488340222317914105737433, 5.26246787718835207711525726175, 6.21853843006083484611157259755, 6.76255511270305335451020935639, 8.102519109866351881401021663630, 8.619340208543478246754406605467, 9.534073902366011424232111164293

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.445063772\)
\(L(\frac12)\)	\(\approx\)	\(1.445063772\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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