LMFDB - L-function 2-12e3-12.11-c3-0-48

Dirichlet series

L(s) = 1

− 1.49i·5^-s − 26.1i·7^-s + 56.3·11^-s + 41.3·13^-s + 51.0i·17^-s + 79.0i·19^-s − 27.3·23^-s + 122.·25^-s + 134. i·29^-s − 187. i·31^-s − 39.1·35^-s + 196.·37^-s + 298. i·41^-s + 465. i·43^-s + 373.·47^-s + ⋯

L(s) = 1

− 0.133i·5^-s − 1.41i·7^-s + 1.54·11^-s + 0.881·13^-s + 0.728i·17^-s + 0.954i·19^-s − 0.248·23^-s + 0.982·25^-s + 0.861i·29^-s − 1.08i·31^-s − 0.189·35^-s + 0.873·37^-s + 1.13i·41^-s + 1.65i·43^-s + 1.16·47^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$1$
Analytic conductor:	$101.955$
Root analytic conductor:	$10.0972$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (1727, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ 1)$

Particular Values

$L(2)$	$\approx$	$2.686986326$
$L(\frac12)$	$\approx$	$2.686986326$
$L(\frac{5}{2})$		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 + 1.49iT - 125T^{2} $
	7	$ 1 + 26.1iT - 343T^{2} $
	11	$ 1 - 56.3T + 1.33e3T^{2} $
	13	$ 1 - 41.3T + 2.19e3T^{2} $
	17	$ 1 - 51.0iT - 4.91e3T^{2} $
	19	$ 1 - 79.0iT - 6.85e3T^{2} $
	23	$ 1 + 27.3T + 1.21e4T^{2} $
	29	$ 1 - 134. iT - 2.43e4T^{2} $
	31	$ 1 + 187. iT - 2.97e4T^{2} $
	37	$ 1 - 196.T + 5.06e4T^{2} $
	41	$ 1 - 298. iT - 6.89e4T^{2} $
	43	$ 1 - 465. iT - 7.95e4T^{2} $
	47	$ 1 - 373.T + 1.03e5T^{2} $
	53	$ 1 - 620. iT - 1.48e5T^{2} $
	59	$ 1 + 321.T + 2.05e5T^{2} $
	61	$ 1 + 674.T + 2.26e5T^{2} $
	67	$ 1 + 576. iT - 3.00e5T^{2} $
	71	$ 1 + 223.T + 3.57e5T^{2} $
	73	$ 1 - 70.1T + 3.89e5T^{2} $
	79	$ 1 - 1.05e3iT - 4.93e5T^{2} $
	83	$ 1 - 1.21e3T + 5.71e5T^{2} $
	89	$ 1 + 1.34e3iT - 7.04e5T^{2} $
	97	$ 1 + 576.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

−9.034005183784751315687987139252, −8.085506608014153822974747233930, −7.43938667066077852231850774874, −6.40134734147361093553425728072, −6.06421972977889406224527414717, −4.49769011466643112848028518006, −4.04446748399460751400120026643, −3.21735351305393628735489215741, −1.46954412504691762454721301419, −0.992774645619208045845945285911, 0.72809569125894633211679568591, 1.95078739028331705739857220041, 2.91072558116383680319985735617, 3.88401663536814898358517320705, 4.92109037391091411838832713540, 5.80911259567354126037077904247, 6.51273678388293944746280901880, 7.21869213831032235832525617160, 8.490963994478124285143249762468, 8.978716139739894126847566132611

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.686986326\)
\(L(\frac12)\)	\(\approx\)	\(2.686986326\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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