LMFDB - L-function 2-7200-24.11-c1-0-15

Dirichlet series

L(s) = 1

− 5.16i·7^-s + 3.05i·11^-s + 0.837i·13^-s − 6.32·19^-s + 4.47·23^-s + 11.1i·37^-s + 10.3i·41^-s − 2.82·47^-s − 19.6·49^-s + 5.65·53^-s + 5.42i·59^-s + 15.7·77^-s − 18.8i·89^-s + 4.32·91^-s + 17.1i·103^-s + ⋯

L(s) = 1

− 1.95i·7^-s + 0.921i·11^-s + 0.232i·13^-s − 1.45·19^-s + 0.932·23^-s + 1.83i·37^-s + 1.61i·41^-s − 0.412·47^-s − 2.80·49^-s + 0.777·53^-s + 0.706i·59^-s + 1.79·77^-s − 1.99i·89^-s + 0.453·91^-s + 1.69i·103^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.577 - 0.816i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (4751, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ 0.577 - 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.312357963$
$L(\frac12)$	$\approx$	$1.312357963$
$L(\frac{3}{2})$		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 + 5.16iT - 7T^{2} $
	11	$ 1 - 3.05iT - 11T^{2} $
	13	$ 1 - 0.837iT - 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 + 6.32T + 19T^{2} $
	23	$ 1 - 4.47T + 23T^{2} $
	29	$ 1 + 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 - 11.1iT - 37T^{2} $
	41	$ 1 - 10.3iT - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 2.82T + 47T^{2} $
	53	$ 1 - 5.65T + 53T^{2} $
	59	$ 1 - 5.42iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 + 18.8iT - 89T^{2} $
	97	$ 1 + 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

−7.928210468216957393285616135706, −7.23786962495907947022901264663, −6.76121264430256393075949065148, −6.19914210641925437247214515479, −4.78697236748583171789862334671, −4.58150537453494949589605880838, −3.81051727675493299330649069809, −2.94403616726886163440773197639, −1.77219510662494249289509476581, −0.941166028461726334426985226107, 0.36002729068997664546460832729, 1.89954678680831273024468555398, 2.51659537270435777626875921046, 3.29594164655683459942272454770, 4.21437149989895313131533259378, 5.30927876959809368755284389195, 5.60134802301789351355991163426, 6.30347594176179982594930619469, 7.01381275013975380452608700618, 8.067078893070165901573545830840

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.312357963\)
\(L(\frac12)\)	\(\approx\)	\(1.312357963\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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