LMFDB - L-function 2-2240-280.139-c1-0-54

Dirichlet series

L(s) = 1

− 2.23·3^-s − 2.23i·5^-s + 2.64i·7^-s + 2.00·9^-s − 5.91·11^-s + 6.70i·13^-s + 5.00i·15^-s + 7.93·17^-s − 5.91i·21^-s − 5.00·25^-s + 2.23·27^-s − 5.91i·29^-s + 13.2·33^-s + 5.91·35^-s − 15.0i·39^-s + ⋯

L(s) = 1

− 1.29·3^-s − 0.999i·5^-s + 0.999i·7^-s + 0.666·9^-s − 1.78·11^-s + 1.86i·13^-s + 1.29i·15^-s + 1.92·17^-s − 1.29i·21^-s − 1.00·25^-s + 0.430·27^-s − 1.09i·29^-s + 2.30·33^-s + 0.999·35^-s − 2.40i·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (1119, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.2220661052$
$L(\frac12)$	$\approx$	$0.2220661052$
$L(\frac{3}{2})$		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 + 2.23iT $
	7	$ 1 - 2.64iT $
good	3	$ 1 + 2.23T + 3T^{2} $
	11	$ 1 + 5.91T + 11T^{2} $
	13	$ 1 - 6.70iT - 13T^{2} $
	17	$ 1 - 7.93T + 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 5.91iT - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 7.93iT - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 + 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 - 12iT - 71T^{2} $
	73	$ 1 + 10.5T + 73T^{2} $
	79	$ 1 + iT - 79T^{2} $
	83	$ 1 + 8.94T + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 + 18.5T + 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

−8.677906104067042810673274648400, −8.067660164833392908763073516263, −7.14403895693631119221110769316, −6.03898618876254211742489442834, −5.51472840612161668740783170211, −5.06219314248462951991628617434, −4.17340726624438098139455270365, −2.69313496649931525044729352145, −1.54771091161204979668360827843, −0.11176235335535481647366422176, 1.00451563547411352772486765651, 2.88239646732095969015657923811, 3.37764216376227579728833995538, 4.83210731628636364830708937046, 5.53038261921122866022856223460, 5.95634435532380918891447828404, 7.08254505534862227758269339743, 7.68185086564779715847281783930, 8.110179462343892239030251846153, 9.867369652765711630797852542139

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2220661052\)
\(L(\frac12)\)	\(\approx\)	\(0.2220661052\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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