LMFDB - L-function 2-180-3.2-c2-0-3

Dirichlet series

L(s) = 1

− 2.23i·5^-s − 13.4·7^-s − 17.6i·11^-s − 7.48·13^-s + 16.9i·17^-s − 10.9·19^-s − 21.9i·23^-s − 5.00·25^-s − 47.3i·29^-s − 16.9·31^-s + 30.1i·35^-s − 5.53·37^-s + 66.3i·41^-s + 38.9·43^-s + 32.5i·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s − 1.92·7^-s − 1.60i·11^-s − 0.575·13^-s + 0.998i·17^-s − 0.577·19^-s − 0.952i·23^-s − 0.200·25^-s − 1.63i·29^-s − 0.547·31^-s + 0.861i·35^-s − 0.149·37^-s + 1.61i·41^-s + 0.906·43^-s + 0.692i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$180$ = $2^{2} \cdot 3^{2} \cdot 5$
Sign:	$-0.816 + 0.577i$
Analytic conductor:	$4.90464$
Root analytic conductor:	$2.21464$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{180} (161, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 180,\ (\ :1),\ -0.816 + 0.577i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.164999 - 0.519131i$
$L(\frac12)$	$\approx$	$0.164999 - 0.519131i$
$L(2)$		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 $
	5	$ 1 + 2.23iT $
good	7	$ 1 + 13.4T + 49T^{2} $
	11	$ 1 + 17.6iT - 121T^{2} $
	13	$ 1 + 7.48T + 169T^{2} $
	17	$ 1 - 16.9iT - 289T^{2} $
	19	$ 1 + 10.9T + 361T^{2} $
	23	$ 1 + 21.9iT - 529T^{2} $
	29	$ 1 + 47.3iT - 841T^{2} $
	31	$ 1 + 16.9T + 961T^{2} $
	37	$ 1 + 5.53T + 1.36e3T^{2} $
	41	$ 1 - 66.3iT - 1.68e3T^{2} $
	43	$ 1 - 38.9T + 1.84e3T^{2} $
	47	$ 1 - 32.5iT - 2.20e3T^{2} $
	53	$ 1 - 11.2iT - 2.80e3T^{2} $
	59	$ 1 + 31.8iT - 3.48e3T^{2} $
	61	$ 1 - 46.9T + 3.72e3T^{2} $
	67	$ 1 + 76T + 4.48e3T^{2} $
	71	$ 1 + 77.7iT - 5.04e3T^{2} $
	73	$ 1 - 94.9T + 5.32e3T^{2} $
	79	$ 1 + 6.92T + 6.24e3T^{2} $
	83	$ 1 + 62.1iT - 6.88e3T^{2} $
	89	$ 1 + 62.2iT - 7.92e3T^{2} $
	97	$ 1 + 124.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

−12.28924252438037121365070466984, −10.96012719497047626991379481162, −9.958034316255844752416700516920, −9.067260749798397705614813280084, −8.062887184178291795752613879799, −6.46754975996789528168434701776, −5.89350633994659501334598119124, −4.05955638928157510233236492774, −2.85437919936228308911185311067, −0.30386543257198405174188844239, 2.50529463217946527320566340079, 3.79106351997745208391470731416, 5.39094811154569674040040368375, 6.90318336016926118128480497204, 7.19075937505196346420280331131, 9.177838239533291693781288482266, 9.762133443206572667084562312414, 10.62776688278013331846767393507, 12.15916709214043497211517272927, 12.66337127236115120627500544014

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.164999 - 0.519131i\)
\(L(\frac12)\)	\(\approx\)	\(0.164999 - 0.519131i\)
\(L(2)\)		not available
\(L(1)\)		not available

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