LMFDB - L-function 2-2240-5.4-c1-0-26

Dirichlet series

L(s) = 1

− 0.363i·3^-s + (1.75 + 1.38i)5^-s − i·7^-s + 2.86·9^-s − 5.14·11^-s + 4.64i·13^-s + (0.504 − 0.636i)15^-s − 3.86i·17^-s − 0.778·19^-s − 0.363·21^-s + 5.00i·23^-s + (1.14 + 4.86i)25^-s − 2.13i·27^-s + 9.42·29^-s + 4.72·31^-s + ⋯

L(s) = 1

− 0.209i·3^-s + (0.783 + 0.621i)5^-s − 0.377i·7^-s + 0.955·9^-s − 1.55·11^-s + 1.28i·13^-s + (0.130 − 0.164i)15^-s − 0.938i·17^-s − 0.178·19^-s − 0.0792·21^-s + 1.04i·23^-s + (0.228 + 0.973i)25^-s − 0.410i·27^-s + 1.74·29^-s + 0.848·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.621 - 0.783i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ 0.621 - 0.783i)$

Particular Values

$L(1)$	$\approx$	$1.951107526$
$L(\frac12)$	$\approx$	$1.951107526$
$L(\frac{3}{2})$		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 + (-1.75 - 1.38i)T $
	7	$ 1 + iT $
good	3	$ 1 + 0.363iT - 3T^{2} $
	11	$ 1 + 5.14T + 11T^{2} $
	13	$ 1 - 4.64iT - 13T^{2} $
	17	$ 1 + 3.86iT - 17T^{2} $
	19	$ 1 + 0.778T + 19T^{2} $
	23	$ 1 - 5.00iT - 23T^{2} $
	29	$ 1 - 9.42T + 29T^{2} $
	31	$ 1 - 4.72T + 31T^{2} $
	37	$ 1 - 6iT - 37T^{2} $
	41	$ 1 + 1.00T + 41T^{2} $
	43	$ 1 - 7.00iT - 43T^{2} $
	47	$ 1 - 11.4iT - 47T^{2} $
	53	$ 1 + 7.55iT - 53T^{2} $
	59	$ 1 - 12.5T + 59T^{2} $
	61	$ 1 + 11.5T + 61T^{2} $
	67	$ 1 + 11.7iT - 67T^{2} $
	71	$ 1 - 2.72T + 71T^{2} $
	73	$ 1 - 5.00iT - 73T^{2} $
	79	$ 1 + 5.68T + 79T^{2} $
	83	$ 1 - 4.67iT - 83T^{2} $
	89	$ 1 - 2.82T + 89T^{2} $
	97	$ 1 - 1.58iT - 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

−9.433406174507939416767765535181, −8.216641807376576950122544900190, −7.50003957166020940662499917226, −6.79542325224046538095816571157, −6.24847097143829099066361432424, −5.04341725020676663427336038811, −4.51098445734869472639692642442, −3.14617134104138740823008942077, −2.35222366684003505636232296208, −1.27035579803350239097332096092, 0.71964500885534532917531131402, 2.10350572550630194825112209802, 2.89715030398171804771310048878, 4.23762783412656626156825203807, 5.03689476391723634347127321189, 5.62725566367529557867879909410, 6.44790611513350791462381533965, 7.49197175266775679555092912510, 8.391374543691498667619725130347, 8.688934370175680591022522687970

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.951107526\)
\(L(\frac12)\)	\(\approx\)	\(1.951107526\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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