LMFDB - L-function 2-7200-40.29-c1-0-15

Dirichlet series

L(s) = 1

+ 1.33i·7^-s + 2.94i·11^-s + 2.04·13^-s + 3.61i·17^-s + 5.35i·19^-s + 8.59i·23^-s + 5.26i·29^-s + 2.08·31^-s + 6.55·37^-s − 7.02·41^-s − 8.50·43^-s − 9.97i·47^-s + 5.22·49^-s − 6.12·53^-s − 4.75i·59^-s + ⋯

L(s) = 1

+ 0.504i·7^-s + 0.887i·11^-s + 0.566·13^-s + 0.876i·17^-s + 1.22i·19^-s + 1.79i·23^-s + 0.977i·29^-s + 0.373·31^-s + 1.07·37^-s − 1.09·41^-s − 1.29·43^-s − 1.45i·47^-s + 0.745·49^-s − 0.841·53^-s − 0.618i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.874 - 0.484i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (2449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ -0.874 - 0.484i)$

Particular Values

$L(1)$	$\approx$	$1.348465103$
$L(\frac12)$	$\approx$	$1.348465103$
$L(\frac{3}{2})$		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 - 1.33iT - 7T^{2} $
	11	$ 1 - 2.94iT - 11T^{2} $
	13	$ 1 - 2.04T + 13T^{2} $
	17	$ 1 - 3.61iT - 17T^{2} $
	19	$ 1 - 5.35iT - 19T^{2} $
	23	$ 1 - 8.59iT - 23T^{2} $
	29	$ 1 - 5.26iT - 29T^{2} $
	31	$ 1 - 2.08T + 31T^{2} $
	37	$ 1 - 6.55T + 37T^{2} $
	41	$ 1 + 7.02T + 41T^{2} $
	43	$ 1 + 8.50T + 43T^{2} $
	47	$ 1 + 9.97iT - 47T^{2} $
	53	$ 1 + 6.12T + 53T^{2} $
	59	$ 1 + 4.75iT - 59T^{2} $
	61	$ 1 + 8.51iT - 61T^{2} $
	67	$ 1 + 10.6T + 67T^{2} $
	71	$ 1 + 2.62T + 71T^{2} $
	73	$ 1 + 15.3iT - 73T^{2} $
	79	$ 1 - 10.4T + 79T^{2} $
	83	$ 1 + 1.52T + 83T^{2} $
	89	$ 1 + 12.7T + 89T^{2} $
	97	$ 1 - 13.4iT - 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

−8.165046515079427856364596449020, −7.62161324753390600628733073561, −6.77544905858596502418340841232, −6.10291488298180161662160815508, −5.43859648783251704499584401834, −4.74572176131480805035257795904, −3.72670771739237775059735305862, −3.28009171400495944331034861774, −1.93427679366342370101674818625, −1.49068784131162374569626402318, 0.34133012771404888550808650973, 1.15110248650026510678433915877, 2.58250807395200471031542364254, 3.03826677997511195327725720735, 4.19304581301405399950220637208, 4.61750470747407078282042653365, 5.57388429170637945146569920509, 6.35262012459428484975158712271, 6.83218603933352804055568053748, 7.65037223377378681794230316495

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.348465103\)
\(L(\frac12)\)	\(\approx\)	\(1.348465103\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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