LMFDB - L-function 2-7200-8.5-c1-0-51

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

$L(1)$	$\approx$	$2.475645915$
$L(\frac12)$	$\approx$	$2.475645915$
$L(\frac{3}{2})$		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

−7.979242712140008503719476405730, −7.53697588255800626156781252856, −6.42121640549334649596324459711, −5.97274207701361946587679312098, −4.99420452955601132959541746932, −4.57267907745349140713655050799, −3.57249496814045865123389087718, −2.81374142585470918815982416314, −1.75110696264098023588771673303, −0.946521519707470345132735664813, 0.76473016138312442922052131849, 1.61500319025012331921068642137, 2.74103741588037917704770001539, 3.44653875530040556789663108003, 4.32428997451345583870910808912, 5.21302051395409203196879527802, 5.56262262009748677517589951754, 6.53083211988350178640373954886, 7.29513692571950158267294368570, 7.83344030362833022624707018091

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.475645915\)
\(L(\frac12)\)	\(\approx\)	\(2.475645915\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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