LMFDB - L-function 2-7200-12.11-c1-0-47

Dirichlet series

L(s) = 1

+ 3.41i·7^-s + 2.58·11^-s + 3.41·13^-s + 1.17i·17^-s − 4.82·23^-s − 6i·29^-s − 6.48i·31^-s + 9.07·37^-s − 11.0i·41^-s − 6.82i·43^-s + 5.65·47^-s − 4.65·49^-s + 1.17i·53^-s + 6.58·59^-s + 12.8·61^-s + ⋯

L(s) = 1

+ 1.29i·7^-s + 0.779·11^-s + 0.946·13^-s + 0.284i·17^-s − 1.00·23^-s − 1.11i·29^-s − 1.16i·31^-s + 1.49·37^-s − 1.72i·41^-s − 1.04i·43^-s + 0.825·47^-s − 0.665·49^-s + 0.160i·53^-s + 0.857·59^-s + 1.64·61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.985 - 0.169i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ 0.985 - 0.169i)$

Particular Values

$L(1)$	$\approx$	$2.266311261$
$L(\frac12)$	$\approx$	$2.266311261$
$L(\frac{3}{2})$		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 - 3.41iT - 7T^{2} $
	11	$ 1 - 2.58T + 11T^{2} $
	13	$ 1 - 3.41T + 13T^{2} $
	17	$ 1 - 1.17iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 4.82T + 23T^{2} $
	29	$ 1 + 6iT - 29T^{2} $
	31	$ 1 + 6.48iT - 31T^{2} $
	37	$ 1 - 9.07T + 37T^{2} $
	41	$ 1 + 11.0iT - 41T^{2} $
	43	$ 1 + 6.82iT - 43T^{2} $
	47	$ 1 - 5.65T + 47T^{2} $
	53	$ 1 - 1.17iT - 53T^{2} $
	59	$ 1 - 6.58T + 59T^{2} $
	61	$ 1 - 12.8T + 61T^{2} $
	67	$ 1 - 8iT - 67T^{2} $
	71	$ 1 - 5.65T + 71T^{2} $
	73	$ 1 - 10.4T + 73T^{2} $
	79	$ 1 + 14.4iT - 79T^{2} $
	83	$ 1 + 9.17T + 83T^{2} $
	89	$ 1 - 4.24iT - 89T^{2} $
	97	$ 1 + 2.48T + 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

−8.155158780131686683633671913977, −7.23947372880354011402440983090, −6.33222637356867366221152544317, −5.88398044791908115405600455634, −5.37073564844382806801997309925, −4.05572360714462264549245999582, −3.85651532060595910431456586441, −2.50548279522217052524061872976, −2.03214573319085963020447460049, −0.73918074386505809380912722733, 0.876522492945685614515021567696, 1.49065456193425446762048180886, 2.81256650270798186955747957127, 3.73190478636747183690248372410, 4.14528097174700802007710805066, 4.98199251839663985825586985149, 5.92849640518488690967904229736, 6.66657262623155798256534034936, 7.03606981433794314792100156374, 8.025959897686537765245484254665

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.266311261\)
\(L(\frac12)\)	\(\approx\)	\(2.266311261\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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