LMFDB - L-function 2-12e3-8.3-c2-0-34

Dirichlet series

L(s) = 1

− 2.64i·5^-s + 8.30i·7^-s − 18.7·11^-s + 22.3i·13^-s − 23.4·17^-s − 9.93·19^-s − 32.3i·23^-s + 18.0·25^-s + 8.45i·29^-s − 46.9i·31^-s + 21.9·35^-s − 28.3i·37^-s + 77.7·41^-s + 58.4·43^-s − 54.2i·47^-s + ⋯

L(s) = 1

− 0.528i·5^-s + 1.18i·7^-s − 1.70·11^-s + 1.71i·13^-s − 1.38·17^-s − 0.522·19^-s − 1.40i·23^-s + 0.721·25^-s + 0.291i·29^-s − 1.51i·31^-s + 0.626·35^-s − 0.765i·37^-s + 1.89·41^-s + 1.35·43^-s − 1.15i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.258 + 0.965i$
Analytic conductor:	$47.0845$
Root analytic conductor:	$6.86182$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1),\ 0.258 + 0.965i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9254520523$
$L(\frac12)$	$\approx$	$0.9254520523$
$L(2)$		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 $
good	5	$ 1 + 2.64iT - 25T^{2} $
	7	$ 1 - 8.30iT - 49T^{2} $
	11	$ 1 + 18.7T + 121T^{2} $
	13	$ 1 - 22.3iT - 169T^{2} $
	17	$ 1 + 23.4T + 289T^{2} $
	19	$ 1 + 9.93T + 361T^{2} $
	23	$ 1 + 32.3iT - 529T^{2} $
	29	$ 1 - 8.45iT - 841T^{2} $
	31	$ 1 + 46.9iT - 961T^{2} $
	37	$ 1 + 28.3iT - 1.36e3T^{2} $
	41	$ 1 - 77.7T + 1.68e3T^{2} $
	43	$ 1 - 58.4T + 1.84e3T^{2} $
	47	$ 1 + 54.2iT - 2.20e3T^{2} $
	53	$ 1 - 5.81iT - 2.80e3T^{2} $
	59	$ 1 + 47.5T + 3.48e3T^{2} $
	61	$ 1 + 27.7iT - 3.72e3T^{2} $
	67	$ 1 - 50.6T + 4.48e3T^{2} $
	71	$ 1 - 7.34iT - 5.04e3T^{2} $
	73	$ 1 + 29.4T + 5.32e3T^{2} $
	79	$ 1 + 43.2iT - 6.24e3T^{2} $
	83	$ 1 + 10.7T + 6.88e3T^{2} $
	89	$ 1 - 136.T + 7.92e3T^{2} $
	97	$ 1 + 54.6T + 9.40e3T^{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

−8.941048960821618820839311788119, −8.417003749447011838902123806138, −7.43314894089042770820387024339, −6.45797613357894294974932339627, −5.73935446030883779196932452599, −4.76130442874651102900458743245, −4.23410928770862697911125759659, −2.38953317849758998056217894360, −2.29698228045456751289608927506, −0.29201619215806631023600027291, 0.899319676586355240772529861537, 2.54119075221654542129724620264, 3.20102609107251885067093809547, 4.35259362727764163621297951405, 5.20386903603064429784580372494, 6.07949414143168107840846250337, 7.12869924892680957785749055630, 7.64936469209651151509527449177, 8.284933232267723073397917414497, 9.397049822765990589087918286520

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9254520523\)
\(L(\frac12)\)	\(\approx\)	\(0.9254520523\)
\(L(2)\)		not available
\(L(1)\)		not available

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