LMFDB - L-function 2-2880-12.11-c1-0-22

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

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

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

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

$L(1)$	$\approx$	$1.873473361$
$L(\frac12)$	$\approx$	$1.873473361$
$L(\frac{3}{2})$		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

−8.573218680345838243915655257234, −7.78431302477789346193336564779, −7.24978720190498031848212027058, −6.20522813531020185721345245495, −5.98882469621893104502891328078, −4.26767265780100038559514460739, −4.14410115823240764998891539091, −3.09562898941084422017937544757, −1.81147926218948905926383692612, −0.68403806485071342146713917172, 1.16545672682995883583708282154, 2.18620615547437263837927221967, 3.25947593760492788234035029807, 4.16102517966306178938739157004, 5.10578831800880141340451404017, 5.88189923629957864994329291679, 6.40041371575159923065587357241, 7.44375135434468858331757518154, 8.345705574839226253370590200004, 8.970492416499444429082742592648

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.873473361\)
\(L(\frac12)\)	\(\approx\)	\(1.873473361\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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