LMFDB - L-function 2-7200-8.5-c1-0-3

Dirichlet series

L(s) = 1

− 4.05·7^-s + 0.985i·11^-s + 4.94i·13^-s + 4.52·17^-s + 2.60i·19^-s − 3.53·23^-s + 7.59i·29^-s + 3.28·31^-s + 0.945i·37^-s − 0.568·41^-s − 8.45i·43^-s − 2.60·47^-s + 9.45·49^-s − 0.229i·53^-s + 9.10i·59^-s + ⋯

L(s) = 1

− 1.53·7^-s + 0.297i·11^-s + 1.37i·13^-s + 1.09·17^-s + 0.597i·19^-s − 0.737·23^-s + 1.41i·29^-s + 0.589·31^-s + 0.155i·37^-s − 0.0887·41^-s − 1.29i·43^-s − 0.379·47^-s + 1.35·49^-s − 0.0315i·53^-s + 1.18i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.973 + 0.227i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (3601, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ -0.973 + 0.227i)$

Particular Values

$L(1)$	$\approx$	$0.3326419783$
$L(\frac12)$	$\approx$	$0.3326419783$
$L(\frac{3}{2})$		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 + 4.05T + 7T^{2} $
	11	$ 1 - 0.985iT - 11T^{2} $
	13	$ 1 - 4.94iT - 13T^{2} $
	17	$ 1 - 4.52T + 17T^{2} $
	19	$ 1 - 2.60iT - 19T^{2} $
	23	$ 1 + 3.53T + 23T^{2} $
	29	$ 1 - 7.59iT - 29T^{2} $
	31	$ 1 - 3.28T + 31T^{2} $
	37	$ 1 - 0.945iT - 37T^{2} $
	41	$ 1 + 0.568T + 41T^{2} $
	43	$ 1 + 8.45iT - 43T^{2} $
	47	$ 1 + 2.60T + 47T^{2} $
	53	$ 1 + 0.229iT - 53T^{2} $
	59	$ 1 - 9.10iT - 59T^{2} $
	61	$ 1 + 11.0iT - 61T^{2} $
	67	$ 1 + 8.45iT - 67T^{2} $
	71	$ 1 - 1.43T + 71T^{2} $
	73	$ 1 + 11.9T + 73T^{2} $
	79	$ 1 - 3.28T + 79T^{2} $
	83	$ 1 - 9.89iT - 83T^{2} $
	89	$ 1 - 12.3T + 89T^{2} $
	97	$ 1 - 3.23T + 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

−8.320516888618133263530974317109, −7.46448417516820406234878246627, −6.82050740824549417575006321956, −6.32164163373813543078136750589, −5.61221597491718140355625261319, −4.72687691906835545792747937728, −3.77235899326316113024246647099, −3.35645102140577605012437337614, −2.32781669329789827229579156772, −1.34302685278608586173527819495, 0.093600023946656762839518189651, 0.994060902680687469153146334524, 2.52917031711116328881831214118, 3.10407842454053201992141621505, 3.72817652328156750975571574432, 4.69405170095547187104470975360, 5.74434445250903457404220114754, 6.00500210595209622202778180299, 6.77713765055788529982652846179, 7.65313576816519351918718714579

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3326419783\)
\(L(\frac12)\)	\(\approx\)	\(0.3326419783\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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