LMFDB - L-function 2-7200-8.5-c1-0-12

Dirichlet series

L(s) = 1

+ 0.0802·7^-s + 2.41i·11^-s + 5.26i·13^-s − 0.255·17^-s + 6.95i·19^-s + 1.64·23^-s + 4.51i·29^-s − 8.29·31^-s + 2.67i·37^-s + 8.11·41^-s + 4.08i·43^-s − 5.70·47^-s − 6.99·49^-s − 11.5i·53^-s − 12.6i·59^-s + ⋯

L(s) = 1

+ 0.0303·7^-s + 0.728i·11^-s + 1.46i·13^-s − 0.0620·17^-s + 1.59i·19^-s + 0.343·23^-s + 0.838i·29^-s − 1.48·31^-s + 0.439i·37^-s + 1.26·41^-s + 0.623i·43^-s − 0.831·47^-s − 0.999·49^-s − 1.58i·53^-s − 1.65i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.990 - 0.136i$
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:	$(2,\ 7200,\ (\ :1/2),\ -0.990 - 0.136i)$

Particular Values

$L(1)$	$\approx$	$0.8436360437$
$L(\frac12)$	$\approx$	$0.8436360437$
$L(\frac{3}{2})$		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 - 0.0802T + 7T^{2} $
	11	$ 1 - 2.41iT - 11T^{2} $
	13	$ 1 - 5.26iT - 13T^{2} $
	17	$ 1 + 0.255T + 17T^{2} $
	19	$ 1 - 6.95iT - 19T^{2} $
	23	$ 1 - 1.64T + 23T^{2} $
	29	$ 1 - 4.51iT - 29T^{2} $
	31	$ 1 + 8.29T + 31T^{2} $
	37	$ 1 - 2.67iT - 37T^{2} $
	41	$ 1 - 8.11T + 41T^{2} $
	43	$ 1 - 4.08iT - 43T^{2} $
	47	$ 1 + 5.70T + 47T^{2} $
	53	$ 1 + 11.5iT - 53T^{2} $
	59	$ 1 + 12.6iT - 59T^{2} $
	61	$ 1 + 11.9iT - 61T^{2} $
	67	$ 1 - 7.27iT - 67T^{2} $
	71	$ 1 + 11.3T + 71T^{2} $
	73	$ 1 - 12.0T + 73T^{2} $
	79	$ 1 + 5.50T + 79T^{2} $
	83	$ 1 + 9.20iT - 83T^{2} $
	89	$ 1 + 11.9T + 89T^{2} $
	97	$ 1 + 8.50T + 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

−8.215268984625537204957048235277, −7.53840218117302506239167565799, −6.82195556094349576680258083941, −6.31301101425836518442191944204, −5.37760897208453247338478986601, −4.70003181107889705728655845493, −3.94344581023576470164595507664, −3.25253618768799978100663067731, −1.97600188551686354053529862991, −1.55128918278656377903905981653, 0.20851816740616069387154272685, 1.13443984574856060602858209684, 2.51794470497285361080117078930, 3.02852084018506243094877463724, 3.95539703098825704364960282027, 4.78739714353881420288083596203, 5.61847832289949751377012017894, 5.98743053482832328297485366590, 7.06076445417383116958970410992, 7.52383110235701490100392896168

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8436360437\)
\(L(\frac12)\)	\(\approx\)	\(0.8436360437\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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