LMFDB - L-function 2-12e3-24.11-c3-0-20

Dirichlet series

L(s) = 1

− 11.9·5^-s − 29.7i·7^-s + 41.2i·11^-s + 17.8·25^-s − 218.·29^-s − 257. i·31^-s + 355. i·35^-s − 540.·49^-s − 247.·53^-s − 493. i·55^-s + 717. i·59^-s − 1.20e3·73^-s + 1.22e3·77^-s + 1.37e3i·79^-s − 1.50e3i·83^-s + ⋯

L(s) = 1

− 1.06·5^-s − 1.60i·7^-s + 1.13i·11^-s + 0.142·25^-s − 1.39·29^-s − 1.49i·31^-s + 1.71i·35^-s − 1.57·49^-s − 0.641·53^-s − 1.20i·55^-s + 1.58i·59^-s − 1.93·73^-s + 1.81·77^-s + 1.95i·79^-s − 1.98i·83^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.707 - 0.707i$
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:	$(2,\ 1728,\ (\ :3/2),\ 0.707 - 0.707i)$

Particular Values

$L(2)$	$\approx$	$0.8674975506$
$L(\frac12)$	$\approx$	$0.8674975506$
$L(\frac{5}{2})$		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 + 11.9T + 125T^{2} $
	7	$ 1 + 29.7iT - 343T^{2} $
	11	$ 1 - 41.2iT - 1.33e3T^{2} $
	13	$ 1 - 2.19e3T^{2} $
	17	$ 1 - 4.91e3T^{2} $
	19	$ 1 + 6.85e3T^{2} $
	23	$ 1 + 1.21e4T^{2} $
	29	$ 1 + 218.T + 2.43e4T^{2} $
	31	$ 1 + 257. iT - 2.97e4T^{2} $
	37	$ 1 - 5.06e4T^{2} $
	41	$ 1 - 6.89e4T^{2} $
	43	$ 1 + 7.95e4T^{2} $
	47	$ 1 + 1.03e5T^{2} $
	53	$ 1 + 247.T + 1.48e5T^{2} $
	59	$ 1 - 717. iT - 2.05e5T^{2} $
	61	$ 1 - 2.26e5T^{2} $
	67	$ 1 + 3.00e5T^{2} $
	71	$ 1 + 3.57e5T^{2} $
	73	$ 1 + 1.20e3T + 3.89e5T^{2} $
	79	$ 1 - 1.37e3iT - 4.93e5T^{2} $
	83	$ 1 + 1.50e3iT - 5.71e5T^{2} $
	89	$ 1 - 7.04e5T^{2} $
	97	$ 1 - 1.86e3T + 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

−9.105625490838983248839565961587, −7.937173714141226219138724474927, −7.46099794053953213699899028363, −7.04547435704609090066637434050, −5.87668420299500770561412238585, −4.54448033168416620010369596882, −4.17807993642523085578042101748, −3.36731455993187816006000070711, −1.91124110912727304608270272615, −0.67501111275389604850478913148, 0.28145028907757604058026380581, 1.80250136947160897355927547856, 3.00608745918441258100371262923, 3.61649119729244734100142165101, 4.83512576174774037167949190550, 5.63390516142589673387268313456, 6.33214397130152473757046694684, 7.42504779927280494105641843815, 8.210108076920627620301709147030, 8.768377524876712925593332547833

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.8674975506\)
\(L(\frac12)\)	\(\approx\)	\(0.8674975506\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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