LMFDB - L-function 2-1120-56.27-c1-0-16

Dirichlet series

L(s) = 1

− 0.528i·3^-s − 5^-s + (2.17 − 1.50i)7^-s + 2.72·9^-s − 3.04·11^-s + 4.75·13^-s + 0.528i·15^-s + 6.78i·17^-s + 0.584i·19^-s + (−0.796 − 1.14i)21^-s − 5.80i·23^-s + 25^-s − 3.02i·27^-s + 0.185i·29^-s + 3.12·31^-s + ⋯

L(s) = 1

− 0.304i·3^-s − 0.447·5^-s + (0.821 − 0.569i)7^-s + 0.907·9^-s − 0.918·11^-s + 1.31·13^-s + 0.136i·15^-s + 1.64i·17^-s + 0.134i·19^-s + (−0.173 − 0.250i)21^-s − 1.21i·23^-s + 0.200·25^-s − 0.581i·27^-s + 0.0344i·29^-s + 0.561·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1120$ = $2^{5} \cdot 5 \cdot 7$
Sign:	$0.810 + 0.585i$
Analytic conductor:	$8.94324$
Root analytic conductor:	$2.99052$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1120} (111, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1120,\ (\ :1/2),\ 0.810 + 0.585i)$

Particular Values

$L(1)$	$\approx$	$1.775465228$
$L(\frac12)$	$\approx$	$1.775465228$
$L(\frac{3}{2})$		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 + (-2.17 + 1.50i)T $
good	3	$ 1 + 0.528iT - 3T^{2} $
	11	$ 1 + 3.04T + 11T^{2} $
	13	$ 1 - 4.75T + 13T^{2} $
	17	$ 1 - 6.78iT - 17T^{2} $
	19	$ 1 - 0.584iT - 19T^{2} $
	23	$ 1 + 5.80iT - 23T^{2} $
	29	$ 1 - 0.185iT - 29T^{2} $
	31	$ 1 - 3.12T + 31T^{2} $
	37	$ 1 + 7.04iT - 37T^{2} $
	41	$ 1 + 3.83iT - 41T^{2} $
	43	$ 1 - 1.43T + 43T^{2} $
	47	$ 1 - 2.95T + 47T^{2} $
	53	$ 1 - 0.535iT - 53T^{2} $
	59	$ 1 - 1.52iT - 59T^{2} $
	61	$ 1 - 13.2T + 61T^{2} $
	67	$ 1 - 9.13T + 67T^{2} $
	71	$ 1 + 9.68iT - 71T^{2} $
	73	$ 1 - 12.4iT - 73T^{2} $
	79	$ 1 + 2.81iT - 79T^{2} $
	83	$ 1 - 13.4iT - 83T^{2} $
	89	$ 1 + 3.10iT - 89T^{2} $
	97	$ 1 + 13.5iT - 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

−9.975358794216561766423322157353, −8.497462510347926420140944646563, −8.233092838422216166463235392539, −7.34332745365088446983894605181, −6.50667534777758761543438282688, −5.49924272848775298135025575574, −4.30577462474064340190220924867, −3.79983820942733046989609521615, −2.15254157518453180757076059161, −0.997187102909741545647440440454, 1.22155211821892572429915485596, 2.66012562213811317430690578927, 3.78416952095104137944463016504, 4.82989425260949892193967022025, 5.39426456290137485670123751556, 6.64624705957000998021518777728, 7.59832584813144202646297245149, 8.193421089185484136445985346046, 9.112329629488584282510519189213, 9.880985267396643682903096993220

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.775465228\)
\(L(\frac12)\)	\(\approx\)	\(1.775465228\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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