LMFDB - L-function 2-1120-40.29-c1-0-6

Dirichlet series

L(s) = 1

− 0.359·3^-s + (−0.565 + 2.16i)5^-s + i·7^-s − 2.87·9^-s − 1.56i·11^-s + 6.61·13^-s + (0.203 − 0.776i)15^-s − 2.81i·17^-s + 5.37i·19^-s − 0.359i·21^-s + 5.85i·23^-s + (−4.35 − 2.44i)25^-s + 2.10·27^-s + 6.75i·29^-s − 9.18·31^-s + ⋯

L(s) = 1

− 0.207·3^-s + (−0.253 + 0.967i)5^-s + 0.377i·7^-s − 0.957·9^-s − 0.472i·11^-s + 1.83·13^-s + (0.0524 − 0.200i)15^-s − 0.683i·17^-s + 1.23i·19^-s − 0.0783i·21^-s + 1.22i·23^-s + (−0.871 − 0.489i)25^-s + 0.405·27^-s + 1.25i·29^-s − 1.65·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1120$ = $2^{5} \cdot 5 \cdot 7$
Sign:	$-0.722 - 0.691i$
Analytic conductor:	$8.94324$
Root analytic conductor:	$2.99052$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1120} (1009, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1120,\ (\ :1/2),\ -0.722 - 0.691i)$

Particular Values

$L(1)$	$\approx$	$0.8757163430$
$L(\frac12)$	$\approx$	$0.8757163430$
$L(\frac{3}{2})$		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 + (0.565 - 2.16i)T $
	7	$ 1 - iT $
good	3	$ 1 + 0.359T + 3T^{2} $
	11	$ 1 + 1.56iT - 11T^{2} $
	13	$ 1 - 6.61T + 13T^{2} $
	17	$ 1 + 2.81iT - 17T^{2} $
	19	$ 1 - 5.37iT - 19T^{2} $
	23	$ 1 - 5.85iT - 23T^{2} $
	29	$ 1 - 6.75iT - 29T^{2} $
	31	$ 1 + 9.18T + 31T^{2} $
	37	$ 1 + 2.55T + 37T^{2} $
	41	$ 1 + 7.93T + 41T^{2} $
	43	$ 1 + 7.16T + 43T^{2} $
	47	$ 1 - 5.24iT - 47T^{2} $
	53	$ 1 + 7.46T + 53T^{2} $
	59	$ 1 - 3.87iT - 59T^{2} $
	61	$ 1 + 4.69iT - 61T^{2} $
	67	$ 1 - 9.92T + 67T^{2} $
	71	$ 1 + 11.4T + 71T^{2} $
	73	$ 1 + 3.67iT - 73T^{2} $
	79	$ 1 - 8.46T + 79T^{2} $
	83	$ 1 - 1.20T + 83T^{2} $
	89	$ 1 + 4.21T + 89T^{2} $
	97	$ 1 - 5.89iT - 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

−10.33568191450295204002817833234, −9.212482232760196388990029832956, −8.501264070596327208933247698886, −7.72930368652780509060983215721, −6.66276823438436934257403697128, −5.93828425089134323791459602614, −5.29187968275223688737160880888, −3.48348223564950824007279438604, −3.34141159330104533734065611367, −1.67691307654704158508774697167, 0.39406293893594653597248430193, 1.80092429690896542234117160235, 3.39344116251364591655932096469, 4.28042854693856046459349418811, 5.21995049902745605439316060659, 6.08815020747453634914063811205, 6.92490631391597724147526250108, 8.244252112378320530984822639683, 8.529855097811738593294439550721, 9.348537393306875867101085591852

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.8757163430\)
\(L(\frac12)\)	\(\approx\)	\(0.8757163430\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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