LMFDB - L-function 2-432-12.11-c3-0-4

Dirichlet series

L(s) = 1

+ 13.3i·5^-s + 19.7i·7^-s + 30.8·11^-s + 72.3·13^-s + 0.430i·17^-s − 22.1i·19^-s − 170.·23^-s − 54.3·25^-s + 269. i·29^-s + 74.4i·31^-s − 264.·35^-s − 287.·37^-s + 136. i·41^-s − 407. i·43^-s + 245.·47^-s + ⋯

L(s) = 1

+ 1.19i·5^-s + 1.06i·7^-s + 0.844·11^-s + 1.54·13^-s + 0.00614i·17^-s − 0.267i·19^-s − 1.54·23^-s − 0.434·25^-s + 1.72i·29^-s + 0.431i·31^-s − 1.27·35^-s − 1.27·37^-s + 0.521i·41^-s − 1.44i·43^-s + 0.762·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$432$ = $2^{4} \cdot 3^{3}$
Sign:	$-0.5 - 0.866i$
Analytic conductor:	$25.4888$
Root analytic conductor:	$5.04864$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{432} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 432,\ (\ :3/2),\ -0.5 - 0.866i)$

Particular Values

$L(2)$	$\approx$	$1.815349995$
$L(\frac12)$	$\approx$	$1.815349995$
$L(\frac{5}{2})$		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 - 13.3iT - 125T^{2} $
	7	$ 1 - 19.7iT - 343T^{2} $
	11	$ 1 - 30.8T + 1.33e3T^{2} $
	13	$ 1 - 72.3T + 2.19e3T^{2} $
	17	$ 1 - 0.430iT - 4.91e3T^{2} $
	19	$ 1 + 22.1iT - 6.85e3T^{2} $
	23	$ 1 + 170.T + 1.21e4T^{2} $
	29	$ 1 - 269. iT - 2.43e4T^{2} $
	31	$ 1 - 74.4iT - 2.97e4T^{2} $
	37	$ 1 + 287.T + 5.06e4T^{2} $
	41	$ 1 - 136. iT - 6.89e4T^{2} $
	43	$ 1 + 407. iT - 7.95e4T^{2} $
	47	$ 1 - 245.T + 1.03e5T^{2} $
	53	$ 1 - 686. iT - 1.48e5T^{2} $
	59	$ 1 + 437.T + 2.05e5T^{2} $
	61	$ 1 - 437.T + 2.26e5T^{2} $
	67	$ 1 + 312. iT - 3.00e5T^{2} $
	71	$ 1 + 356.T + 3.57e5T^{2} $
	73	$ 1 - 434.T + 3.89e5T^{2} $
	79	$ 1 - 544. iT - 4.93e5T^{2} $
	83	$ 1 + 1.42e3T + 5.71e5T^{2} $
	89	$ 1 + 21.6iT - 7.04e5T^{2} $
	97	$ 1 - 273.T + 9.12e5T^{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

−10.97884057171252211640297419687, −10.33545953737909227681337508522, −9.056300154634384570233288502040, −8.516064160990477197993142893719, −7.14317884364119441014960310117, −6.34786841723425157622980565996, −5.57370510739316837923240940092, −3.91818132539597465826074870813, −2.99088620039862275517369946625, −1.65794004840322212209700559744, 0.62534123095145325551838549072, 1.63223349127180174178892047862, 3.80233077029571880167730659434, 4.28545963183724762223200672371, 5.70064824001362034633836133367, 6.58186570410277785892047426363, 7.889928389690569161061528416826, 8.544577483100393458356325557501, 9.505995274925178147636535057622, 10.37368441757143549655905095046

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 $\F_{q}$
Belyi maps

\(L(2)\)	\(\approx\)	\(1.815349995\)
\(L(\frac12)\)	\(\approx\)	\(1.815349995\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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