LMFDB - L-function 2-12e3-24.11-c3-0-22

Dirichlet series

L(s) = 1

− 12.9·5^-s − 31.5i·7^-s − 13.9i·11^-s − 13.5i·13^-s + 123. i·17^-s − 147.·19^-s − 198.·23^-s + 41.6·25^-s + 189.·29^-s + 280. i·31^-s + 407. i·35^-s − 348. i·37^-s + 285. i·41^-s + 61.6·43^-s + 137.·47^-s + ⋯

L(s) = 1

− 1.15·5^-s − 1.70i·7^-s − 0.383i·11^-s − 0.289i·13^-s + 1.75i·17^-s − 1.77·19^-s − 1.80·23^-s + 0.333·25^-s + 1.21·29^-s + 1.62i·31^-s + 1.96i·35^-s − 1.54i·37^-s + 1.08i·41^-s + 0.218·43^-s + 0.427·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.965 - 0.258i$
Analytic conductor:	$101.955$
Root analytic conductor:	$10.0972$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (863, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ 0.965 - 0.258i)$

Particular Values

$L(2)$	$\approx$	$0.7739109690$
$L(\frac12)$	$\approx$	$0.7739109690$
$L(\frac{5}{2})$		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 + 12.9T + 125T^{2} $
	7	$ 1 + 31.5iT - 343T^{2} $
	11	$ 1 + 13.9iT - 1.33e3T^{2} $
	13	$ 1 + 13.5iT - 2.19e3T^{2} $
	17	$ 1 - 123. iT - 4.91e3T^{2} $
	19	$ 1 + 147.T + 6.85e3T^{2} $
	23	$ 1 + 198.T + 1.21e4T^{2} $
	29	$ 1 - 189.T + 2.43e4T^{2} $
	31	$ 1 - 280. iT - 2.97e4T^{2} $
	37	$ 1 + 348. iT - 5.06e4T^{2} $
	41	$ 1 - 285. iT - 6.89e4T^{2} $
	43	$ 1 - 61.6T + 7.95e4T^{2} $
	47	$ 1 - 137.T + 1.03e5T^{2} $
	53	$ 1 - 418.T + 1.48e5T^{2} $
	59	$ 1 + 567. iT - 2.05e5T^{2} $
	61	$ 1 - 286. iT - 2.26e5T^{2} $
	67	$ 1 + 784.T + 3.00e5T^{2} $
	71	$ 1 + 357.T + 3.57e5T^{2} $
	73	$ 1 - 87.3T + 3.89e5T^{2} $
	79	$ 1 + 539. iT - 4.93e5T^{2} $
	83	$ 1 + 365. iT - 5.71e5T^{2} $
	89	$ 1 - 626. iT - 7.04e5T^{2} $
	97	$ 1 - 789.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

−8.615241810921291403833214204075, −8.192607880579070844768121124748, −7.53586082140242375530050278674, −6.66306735561370179337704487180, −5.95601263458750820922262710871, −4.35190288566403582406226884345, −4.14517400469707078294770255441, −3.34853842367133767167591603564, −1.76602054119968281375722730021, −0.55043305150120490673025026558, 0.29728303517243598286505834966, 2.12627638151765409600581578765, 2.72769914363670787480065864565, 4.05678992926913396110928193067, 4.67034512091626817447612581008, 5.74215255883343709473089721177, 6.47483143586154731791831161892, 7.46721352769144765356660195661, 8.243344890882985615799366251490, 8.780453939197823508385582857629

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

\(L(2)\)	\(\approx\)	\(0.7739109690\)
\(L(\frac12)\)	\(\approx\)	\(0.7739109690\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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