LMFDB - L-function 2-2880-12.11-c1-0-23

Dirichlet series

L(s) = 1

− i·5^-s + 0.585i·7^-s + 5.41·11^-s + 0.585·13^-s − 6.82i·17^-s + 0.828·23^-s − 25^-s + 6i·29^-s − 10.4i·31^-s + 0.585·35^-s − 5.07·37^-s + 3.07i·41^-s + 1.17i·43^-s − 5.65·47^-s + 6.65·49^-s + ⋯

L(s) = 1

− 0.447i·5^-s + 0.221i·7^-s + 1.63·11^-s + 0.162·13^-s − 1.65i·17^-s + 0.172·23^-s − 0.200·25^-s + 1.11i·29^-s − 1.88i·31^-s + 0.0990·35^-s − 0.833·37^-s + 0.479i·41^-s + 0.178i·43^-s − 0.825·47^-s + 0.950·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2880$ = $2^{6} \cdot 3^{2} \cdot 5$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$22.9969$
Root analytic conductor:	$4.79550$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2880} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2880,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.956512084$
$L(\frac12)$	$\approx$	$1.956512084$
$L(\frac{3}{2})$		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 - 0.585iT - 7T^{2} $
	11	$ 1 - 5.41T + 11T^{2} $
	13	$ 1 - 0.585T + 13T^{2} $
	17	$ 1 + 6.82iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 - 0.828T + 23T^{2} $
	29	$ 1 - 6iT - 29T^{2} $
	31	$ 1 + 10.4iT - 31T^{2} $
	37	$ 1 + 5.07T + 37T^{2} $
	41	$ 1 - 3.07iT - 41T^{2} $
	43	$ 1 - 1.17iT - 43T^{2} $
	47	$ 1 + 5.65T + 47T^{2} $
	53	$ 1 - 6.82iT - 53T^{2} $
	59	$ 1 - 9.41T + 59T^{2} $
	61	$ 1 + 7.17T + 61T^{2} $
	67	$ 1 + 8iT - 67T^{2} $
	71	$ 1 - 5.65T + 71T^{2} $
	73	$ 1 - 6.48T + 73T^{2} $
	79	$ 1 + 2.48iT - 79T^{2} $
	83	$ 1 - 14.8T + 83T^{2} $
	89	$ 1 + 4.24iT - 89T^{2} $
	97	$ 1 + 14.4T + 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

−8.894671700058966766610472494744, −7.916750833219756058746316214702, −7.08703602526876294146366899596, −6.44341221351977276953438637316, −5.55291599074934418575852842752, −4.74104378048026858674767721501, −3.94710090700607479079461626341, −3.00980662863888316510312394393, −1.81105873363185615404222115803, −0.71854391086538138962759491540, 1.17775957679559021096934900490, 2.12791163095835173528975511613, 3.58764856693643024374507862748, 3.84660638533186674130187240649, 4.98891795800323230134730111502, 6.05970324140307381309994069260, 6.58807887239769509701043577513, 7.21809004824463683741692450768, 8.304721768956412644874100101711, 8.762439034450376448817158038063

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.956512084\)
\(L(\frac12)\)	\(\approx\)	\(1.956512084\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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