LMFDB - L-function 2-7200-8.5-c1-0-29

Dirichlet series

L(s) = 1

− 1.97·7^-s + 1.43i·11^-s + 0.241i·13^-s − 7.38·17^-s + 3.04i·19^-s − 0.874·23^-s − 9.07i·29^-s + 7.44·31^-s − 8.81i·37^-s + 1.91·41^-s − 11.2i·43^-s + 3.34·47^-s − 3.09·49^-s + 9.20i·53^-s + 6.43i·59^-s + ⋯

L(s) = 1

− 0.747·7^-s + 0.431i·11^-s + 0.0669i·13^-s − 1.79·17^-s + 0.697i·19^-s − 0.182·23^-s − 1.68i·29^-s + 1.33·31^-s − 1.44i·37^-s + 0.298·41^-s − 1.71i·43^-s + 0.487·47^-s − 0.441·49^-s + 1.26i·53^-s + 0.837i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.913 - 0.407i$
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:	$(2,\ 7200,\ (\ :1/2),\ 0.913 - 0.407i)$

Particular Values

$L(1)$	$\approx$	$1.305186441$
$L(\frac12)$	$\approx$	$1.305186441$
$L(\frac{3}{2})$		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 + 1.97T + 7T^{2} $
	11	$ 1 - 1.43iT - 11T^{2} $
	13	$ 1 - 0.241iT - 13T^{2} $
	17	$ 1 + 7.38T + 17T^{2} $
	19	$ 1 - 3.04iT - 19T^{2} $
	23	$ 1 + 0.874T + 23T^{2} $
	29	$ 1 + 9.07iT - 29T^{2} $
	31	$ 1 - 7.44T + 31T^{2} $
	37	$ 1 + 8.81iT - 37T^{2} $
	41	$ 1 - 1.91T + 41T^{2} $
	43	$ 1 + 11.2iT - 43T^{2} $
	47	$ 1 - 3.34T + 47T^{2} $
	53	$ 1 - 9.20iT - 53T^{2} $
	59	$ 1 - 6.43iT - 59T^{2} $
	61	$ 1 + 4.57iT - 61T^{2} $
	67	$ 1 - 4.86iT - 67T^{2} $
	71	$ 1 + 8.21T + 71T^{2} $
	73	$ 1 - 4.12T + 73T^{2} $
	79	$ 1 - 13.6T + 79T^{2} $
	83	$ 1 - 12.3iT - 83T^{2} $
	89	$ 1 - 8.08T + 89T^{2} $
	97	$ 1 + 10.6T + 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

−7.87945094001683544120013634189, −7.29011947817155165240609140874, −6.40892109526764476640589265867, −6.13543773945576376821087278415, −5.12615441792352694753914644246, −4.22126734443430116737667530084, −3.82211211865762428217879009677, −2.57177943046635331029932758941, −2.10146928757475369297659557947, −0.62899057023146097729355336372, 0.49818267585777766670505006771, 1.75342076538122193436632351550, 2.84248313181945500106284613810, 3.29583554416496490142647763935, 4.47144764029853572856971055216, 4.85532348295227620977208497406, 5.95050227671182832656755813365, 6.61606542167049889004391124517, 6.89335912816126957127089261203, 7.986650543151018722689276966330

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.305186441\)
\(L(\frac12)\)	\(\approx\)	\(1.305186441\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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