LMFDB - L-function 2-7200-60.59-c1-0-41

Dirichlet series

L(s) = 1

− 3.41·7^-s − 2.58·11^-s + 3.41i·13^-s + 1.17·17^-s + 4.82i·23^-s + 6i·29^-s + 6.48i·31^-s − 9.07i·37^-s − 11.0i·41^-s − 6.82·43^-s + 5.65i·47^-s + 4.65·49^-s − 1.17·53^-s + 6.58·59^-s + 12.8·61^-s + ⋯

L(s) = 1

− 1.29·7^-s − 0.779·11^-s + 0.946i·13^-s + 0.284·17^-s + 1.00i·23^-s + 1.11i·29^-s + 1.16i·31^-s − 1.49i·37^-s − 1.72i·41^-s − 1.04·43^-s + 0.825i·47^-s + 0.665·49^-s − 0.160·53^-s + 0.857·59^-s + 1.64·61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.289 + 0.957i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (7199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ -0.289 + 0.957i)$

Particular Values

$L(1)$	$\approx$	$0.4031635136$
$L(\frac12)$	$\approx$	$0.4031635136$
$L(\frac{3}{2})$		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 $
good	7	$ 1 + 3.41T + 7T^{2} $
	11	$ 1 + 2.58T + 11T^{2} $
	13	$ 1 - 3.41iT - 13T^{2} $
	17	$ 1 - 1.17T + 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 - 4.82iT - 23T^{2} $
	29	$ 1 - 6iT - 29T^{2} $
	31	$ 1 - 6.48iT - 31T^{2} $
	37	$ 1 + 9.07iT - 37T^{2} $
	41	$ 1 + 11.0iT - 41T^{2} $
	43	$ 1 + 6.82T + 43T^{2} $
	47	$ 1 - 5.65iT - 47T^{2} $
	53	$ 1 + 1.17T + 53T^{2} $
	59	$ 1 - 6.58T + 59T^{2} $
	61	$ 1 - 12.8T + 61T^{2} $
	67	$ 1 + 8T + 67T^{2} $
	71	$ 1 + 5.65T + 71T^{2} $
	73	$ 1 - 10.4iT - 73T^{2} $
	79	$ 1 + 14.4iT - 79T^{2} $
	83	$ 1 - 9.17iT - 83T^{2} $
	89	$ 1 + 4.24iT - 89T^{2} $
	97	$ 1 - 2.48iT - 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

−7.43624052044888195397069794010, −7.12272522169466146391245002730, −6.37796826255552673629989829318, −5.58827842360510410042799300938, −5.02614859713352584923845810512, −3.90854125550556522160460855502, −3.37944928818316344147309269836, −2.53848758468994478253275343742, −1.53157596906930454494637877286, −0.12371436441068467045418841298, 0.815334436864644077930217430145, 2.33132675102673531319378318618, 2.96242257480360393066443252093, 3.63724946069284620817362666859, 4.62185788831199352584417306270, 5.35531169134511589086403493593, 6.21168610103258061444810050343, 6.52154445500981819886992963709, 7.53399343130081352573984404796, 8.090068835700690227173240431967

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4031635136\)
\(L(\frac12)\)	\(\approx\)	\(0.4031635136\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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