LMFDB - L-function 2-1440-12.11-c1-0-12

Dirichlet series

L(s) = 1

+ i·5^-s − 3.41i·7^-s + 2.58·11^-s − 3.41·13^-s − 1.17i·17^-s + 4.82·23^-s − 25^-s − 6i·29^-s − 6.48i·31^-s + 3.41·35^-s − 9.07·37^-s − 11.0i·41^-s + 6.82i·43^-s − 5.65·47^-s − 4.65·49^-s + ⋯

L(s) = 1

+ 0.447i·5^-s − 1.29i·7^-s + 0.779·11^-s − 0.946·13^-s − 0.284i·17^-s + 1.00·23^-s − 0.200·25^-s − 1.11i·29^-s − 1.16i·31^-s + 0.577·35^-s − 1.49·37^-s − 1.72i·41^-s + 1.04i·43^-s − 0.825·47^-s − 0.665·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1440$ = $2^{5} \cdot 3^{2} \cdot 5$
Sign:	$0.169 + 0.985i$
Analytic conductor:	$11.4984$
Root analytic conductor:	$3.39093$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1440} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1440,\ (\ :1/2),\ 0.169 + 0.985i)$

Particular Values

$L(1)$	$\approx$	$1.406936740$
$L(\frac12)$	$\approx$	$1.406936740$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	5	$ 1 - iT $
good	7	$ 1 + 3.41iT - 7T^{2} $
	11	$ 1 - 2.58T + 11T^{2} $
	13	$ 1 + 3.41T + 13T^{2} $
	17	$ 1 + 1.17iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 - 4.82T + 23T^{2} $
	29	$ 1 + 6iT - 29T^{2} $
	31	$ 1 + 6.48iT - 31T^{2} $
	37	$ 1 + 9.07T + 37T^{2} $
	41	$ 1 + 11.0iT - 41T^{2} $
	43	$ 1 - 6.82iT - 43T^{2} $
	47	$ 1 + 5.65T + 47T^{2} $
	53	$ 1 + 1.17iT - 53T^{2} $
	59	$ 1 - 6.58T + 59T^{2} $
	61	$ 1 - 12.8T + 61T^{2} $
	67	$ 1 + 8iT - 67T^{2} $
	71	$ 1 - 5.65T + 71T^{2} $
	73	$ 1 + 10.4T + 73T^{2} $
	79	$ 1 + 14.4iT - 79T^{2} $
	83	$ 1 - 9.17T + 83T^{2} $
	89	$ 1 - 4.24iT - 89T^{2} $
	97	$ 1 - 2.48T + 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

−9.566050017767219109662515145369, −8.537446593605252627923973779353, −7.44157884565569362876158687670, −7.11164819831767415501807486801, −6.24144505599855052508600597378, −5.05890682396655629867151898706, −4.16509758602612976716246885438, −3.36385352000344605847600903765, −2.08848312519815980533720946899, −0.58624250606650532635251091365, 1.42931743693367501213668590270, 2.58393788639444131143752976573, 3.62749584222220883590477344941, 4.97203497145549310428126572491, 5.32987622172060989172101294731, 6.51000011221042397013580553201, 7.16499808477412468660353782763, 8.435108658306356957114319435739, 8.817531194035190798980685073522, 9.548678562264761436604187850987

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.406936740\)
\(L(\frac12)\)	\(\approx\)	\(1.406936740\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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