LMFDB - L-function 2-8280-69.68-c1-0-15

Dirichlet series

L(s) = 1

− 5^-s + 2.09i·7^-s − 3.07·11^-s + 3.07·13^-s + 2.79·17^-s + 7.09i·19^-s + (−2.07 − 4.32i)23^-s + 25^-s + 7.02i·29^-s + 2.54·31^-s − 2.09i·35^-s + 11.8i·37^-s + 1.34i·41^-s − 7.71i·43^-s + 2.64i·47^-s + ⋯

L(s) = 1

− 0.447·5^-s + 0.790i·7^-s − 0.927·11^-s + 0.852·13^-s + 0.678·17^-s + 1.62i·19^-s + (−0.433 − 0.901i)23^-s + 0.200·25^-s + 1.30i·29^-s + 0.457·31^-s − 0.353i·35^-s + 1.94i·37^-s + 0.210i·41^-s − 1.17i·43^-s + 0.385i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8280$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$-0.874 - 0.485i$
Analytic conductor:	$66.1161$
Root analytic conductor:	$8.13118$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8280} (1241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8280,\ (\ :1/2),\ -0.874 - 0.485i)$

Particular Values

$L(1)$	$\approx$	$0.9616171633$
$L(\frac12)$	$\approx$	$0.9616171633$
$L(\frac{3}{2})$		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 + T $
	23	$ 1 + (2.07 + 4.32i)T $
good	7	$ 1 - 2.09iT - 7T^{2} $
	11	$ 1 + 3.07T + 11T^{2} $
	13	$ 1 - 3.07T + 13T^{2} $
	17	$ 1 - 2.79T + 17T^{2} $
	19	$ 1 - 7.09iT - 19T^{2} $
	29	$ 1 - 7.02iT - 29T^{2} $
	31	$ 1 - 2.54T + 31T^{2} $
	37	$ 1 - 11.8iT - 37T^{2} $
	41	$ 1 - 1.34iT - 41T^{2} $
	43	$ 1 + 7.71iT - 43T^{2} $
	47	$ 1 - 2.64iT - 47T^{2} $
	53	$ 1 + 5.21T + 53T^{2} $
	59	$ 1 + 9.91iT - 59T^{2} $
	61	$ 1 + 7.34iT - 61T^{2} $
	67	$ 1 - 1.63iT - 67T^{2} $
	71	$ 1 + 13.2iT - 71T^{2} $
	73	$ 1 + 10.0T + 73T^{2} $
	79	$ 1 + 0.0188iT - 79T^{2} $
	83	$ 1 + 1.02T + 83T^{2} $
	89	$ 1 - 8.49T + 89T^{2} $
	97	$ 1 - 12.5iT - 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

−8.111949081692438426795019388441, −7.65111298342315820850153788298, −6.55102788431675691855465083204, −6.08160445836699647319727982907, −5.29185289063058645192322612481, −4.71956047077133017974032363787, −3.62727600546867799757049291037, −3.16550252159698166024423999196, −2.15620391730952091073203609004, −1.20223291951995248385080572054, 0.25180671237432304505926207074, 1.14891521029808481672064945478, 2.41449254150863562946782141760, 3.17749371567390299820658087109, 4.04591400463718561709167494443, 4.52848982514008961586198262111, 5.53405332028860681344129310162, 6.02590633248267467721839992340, 7.11200855416169562584567144615, 7.43613244864510506799599326643

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9616171633\)
\(L(\frac12)\)	\(\approx\)	\(0.9616171633\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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