LMFDB - L-function 2-7200-8.5-c1-0-73

Dirichlet series

L(s) = 1

+ 4.68·7^-s − 2.29i·11^-s − 4.97i·13^-s − 2.97·17^-s + 2.68i·19^-s − 2.68·23^-s + 2i·29^-s + 6.97·31^-s + 4.39i·37^-s + 11.3·41^-s − 9.37i·43^-s − 7.27·47^-s + 14.9·49^-s + 2i·53^-s − 1.70i·59^-s + ⋯

L(s) = 1

+ 1.77·7^-s − 0.691i·11^-s − 1.38i·13^-s − 0.722·17^-s + 0.616i·19^-s − 0.560·23^-s + 0.371i·29^-s + 1.25·31^-s + 0.722i·37^-s + 1.77·41^-s − 1.42i·43^-s − 1.06·47^-s + 2.13·49^-s + 0.274i·53^-s − 0.222i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.474 + 0.880i$
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:	$(2,\ 7200,\ (\ :1/2),\ 0.474 + 0.880i)$

Particular Values

$L(1)$	$\approx$	$2.429800362$
$L(\frac12)$	$\approx$	$2.429800362$
$L(\frac{3}{2})$		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 - 4.68T + 7T^{2} $
	11	$ 1 + 2.29iT - 11T^{2} $
	13	$ 1 + 4.97iT - 13T^{2} $
	17	$ 1 + 2.97T + 17T^{2} $
	19	$ 1 - 2.68iT - 19T^{2} $
	23	$ 1 + 2.68T + 23T^{2} $
	29	$ 1 - 2iT - 29T^{2} $
	31	$ 1 - 6.97T + 31T^{2} $
	37	$ 1 - 4.39iT - 37T^{2} $
	41	$ 1 - 11.3T + 41T^{2} $
	43	$ 1 + 9.37iT - 43T^{2} $
	47	$ 1 + 7.27T + 47T^{2} $
	53	$ 1 - 2iT - 53T^{2} $
	59	$ 1 + 1.70iT - 59T^{2} $
	61	$ 1 + 4.58iT - 61T^{2} $
	67	$ 1 + 4iT - 67T^{2} $
	71	$ 1 - 0.585T + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 + 1.02T + 79T^{2} $
	83	$ 1 + 13.3iT - 83T^{2} $
	89	$ 1 + 3.37T + 89T^{2} $
	97	$ 1 - 3.95T + 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

−8.032784388407818186942610202370, −7.30914051831331773274825956889, −6.26520592619022630210060914567, −5.65159476088932019835613760575, −4.96721732401847780418958799355, −4.35058809302735233398783053479, −3.42042572764493985305555754546, −2.49457865670244400671384909813, −1.59882775035723818790569833803, −0.63448110317034944870442344453, 1.11394030276598942753880896861, 2.00411685406739795977869107082, 2.53616866702306468831509828492, 4.09242241733019606174409759864, 4.46983270564932878369861118200, 4.97323394342446369688736042280, 5.98033056477162817943216248535, 6.72696632802560436272707098217, 7.40445288657100967733092253722, 8.047498767206225585648424001438

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.429800362\)
\(L(\frac12)\)	\(\approx\)	\(2.429800362\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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