LMFDB - L-function 2-2240-56.27-c1-0-47

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$-0.205 + 0.978i$
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:	$(2,\ 2240,\ (\ :1/2),\ -0.205 + 0.978i)$

Particular Values

$L(1)$	$\approx$	$2.263325235$
$L(\frac12)$	$\approx$	$2.263325235$
$L(\frac{3}{2})$		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

−8.599761752790563779559534982308, −8.008229339205557605802512744782, −7.28023882187444131224522858235, −6.54297198584778588589309219159, −5.77650403427805886474403701030, −5.19842442506973239257263779199, −3.76146558956703700609299170651, −2.52198614855643066742766163124, −1.76549978735500967672207767205, −0.912805116750998067070415614547, 1.27631137370010869661123298143, 2.72920422366590984603972020529, 3.78406163707568178444554676698, 4.40949663338103074470423609733, 5.04777437569547879569490785827, 5.97446667853924921213898671889, 6.71117284541603243934365680707, 8.239699959009001733924904047136, 8.408572963601150447759503950489, 9.444502531138645312977887818370

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.263325235\)
\(L(\frac12)\)	\(\approx\)	\(2.263325235\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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