LMFDB - L-function 2-12e3-24.5-c2-0-38

Dirichlet series

L(s) = 1

− 3.32·5^-s + 4.21·7^-s + 16.0·11^-s − 17.6i·13^-s + 4.44i·17^-s + 26.5i·19^-s + 5.59i·23^-s − 13.9·25^-s + 50.0·29^-s − 11.4·31^-s − 14.0·35^-s + 24.5i·37^-s − 18.5i·41^-s − 37.2i·43^-s − 73.5i·47^-s + ⋯

L(s) = 1

− 0.664·5^-s + 0.602·7^-s + 1.46·11^-s − 1.35i·13^-s + 0.261i·17^-s + 1.39i·19^-s + 0.243i·23^-s − 0.558·25^-s + 1.72·29^-s − 0.370·31^-s − 0.400·35^-s + 0.664i·37^-s − 0.453i·41^-s − 0.865i·43^-s − 1.56i·47^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.965 + 0.258i$
Analytic conductor:	$47.0845$
Root analytic conductor:	$6.86182$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (161, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1),\ 0.965 + 0.258i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.065674626$
$L(\frac12)$	$\approx$	$2.065674626$
$L(2)$		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 + 3.32T + 25T^{2} $
	7	$ 1 - 4.21T + 49T^{2} $
	11	$ 1 - 16.0T + 121T^{2} $
	13	$ 1 + 17.6iT - 169T^{2} $
	17	$ 1 - 4.44iT - 289T^{2} $
	19	$ 1 - 26.5iT - 361T^{2} $
	23	$ 1 - 5.59iT - 529T^{2} $
	29	$ 1 - 50.0T + 841T^{2} $
	31	$ 1 + 11.4T + 961T^{2} $
	37	$ 1 - 24.5iT - 1.36e3T^{2} $
	41	$ 1 + 18.5iT - 1.68e3T^{2} $
	43	$ 1 + 37.2iT - 1.84e3T^{2} $
	47	$ 1 + 73.5iT - 2.20e3T^{2} $
	53	$ 1 - 2.32T + 2.80e3T^{2} $
	59	$ 1 - 26.8T + 3.48e3T^{2} $
	61	$ 1 - 12.3iT - 3.72e3T^{2} $
	67	$ 1 - 45.2iT - 4.48e3T^{2} $
	71	$ 1 + 83.3iT - 5.04e3T^{2} $
	73	$ 1 + 3.04T + 5.32e3T^{2} $
	79	$ 1 - 17.2T + 6.24e3T^{2} $
	83	$ 1 - 92.8T + 6.88e3T^{2} $
	89	$ 1 - 160. iT - 7.92e3T^{2} $
	97	$ 1 - 122.T + 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

−8.921689064093242867119907510631, −8.224840634164339421205768953810, −7.72259212180456214637142125465, −6.71561903203856605898224911034, −5.88498236430196378486255604145, −4.97446127782919726590294189366, −3.93202214508891987266270111242, −3.38124343092367607996745227319, −1.86713152290127601335912686592, −0.77355368633971625700164765701, 0.879653954104533514376895772608, 2.01518220198001312007553521524, 3.29169597355637141740948798298, 4.45442365039343894255871241638, 4.61464017738884294923404237616, 6.16968973563668195767390551906, 6.78694180788451514468974174406, 7.53223260139957217204398192955, 8.489121597930310417340476402942, 9.106026142402311083628843970440

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.065674626\)
\(L(\frac12)\)	\(\approx\)	\(2.065674626\)
\(L(2)\)		not available
\(L(1)\)		not available

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