LMFDB - L-function 2-2240-56.27-c1-0-7

Dirichlet series

L(s) = 1

− 2.91i·3^-s − 5^-s + (2.36 − 1.19i)7^-s − 5.50·9^-s − 0.723·11^-s − 4.05·13^-s + 2.91i·15^-s + 6.74i·17^-s + 8.55i·19^-s + (−3.48 − 6.88i)21^-s + 2.79i·23^-s + 25^-s + 7.30i·27^-s + 6.62i·29^-s + 4.51·31^-s + ⋯

L(s) = 1

− 1.68i·3^-s − 0.447·5^-s + (0.892 − 0.451i)7^-s − 1.83·9^-s − 0.218·11^-s − 1.12·13^-s + 0.752i·15^-s + 1.63i·17^-s + 1.96i·19^-s + (−0.760 − 1.50i)21^-s + 0.583i·23^-s + 0.200·25^-s + 1.40i·27^-s + 1.23i·29^-s + 0.811·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.745 - 0.667i$
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:	$(2,\ 2240,\ (\ :1/2),\ 0.745 - 0.667i)$

Particular Values

$L(1)$	$\approx$	$0.7614016491$
$L(\frac12)$	$\approx$	$0.7614016491$
$L(\frac{3}{2})$		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 + (-2.36 + 1.19i)T $
good	3	$ 1 + 2.91iT - 3T^{2} $
	11	$ 1 + 0.723T + 11T^{2} $
	13	$ 1 + 4.05T + 13T^{2} $
	17	$ 1 - 6.74iT - 17T^{2} $
	19	$ 1 - 8.55iT - 19T^{2} $
	23	$ 1 - 2.79iT - 23T^{2} $
	29	$ 1 - 6.62iT - 29T^{2} $
	31	$ 1 - 4.51T + 31T^{2} $
	37	$ 1 + 10.4iT - 37T^{2} $
	41	$ 1 - 9.91iT - 41T^{2} $
	43	$ 1 + 5.50T + 43T^{2} $
	47	$ 1 + 7.93T + 47T^{2} $
	53	$ 1 + 5.97iT - 53T^{2} $
	59	$ 1 - 6.56iT - 59T^{2} $
	61	$ 1 + 13.0T + 61T^{2} $
	67	$ 1 + 5.58T + 67T^{2} $
	71	$ 1 - 3.78iT - 71T^{2} $
	73	$ 1 - 7.79iT - 73T^{2} $
	79	$ 1 - 3.02iT - 79T^{2} $
	83	$ 1 + 4.62iT - 83T^{2} $
	89	$ 1 + 15.4iT - 89T^{2} $
	97	$ 1 - 1.26iT - 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

−8.691850071406991994487660757424, −7.945192571366473011067944425361, −7.78898122068190869043910179820, −6.97732860199428717419083024729, −6.13815981843308063678523309912, −5.35736258844270195194492549292, −4.26042440847559080031096402676, −3.19346651029688025246925541323, −1.87346915778199354755388471457, −1.37864713347889795334804669742, 0.26044319716686123687821217357, 2.52044797951275066484860980351, 3.06041541208668769478460067332, 4.52818468187899144905979690854, 4.73290123501124105601481512191, 5.27005376322328598065044938052, 6.59785925367354002449712608065, 7.56739208904290292447747767003, 8.352078659416760661356544668613, 9.164979742798620436350717818901

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7614016491\)
\(L(\frac12)\)	\(\approx\)	\(0.7614016491\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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