LMFDB - L-function 2-12e3-8.5-c3-0-72

Dirichlet series

L(s) = 1

− 6.50i·5^-s − 16.0·7^-s − 27.9i·11^-s − 35.5i·13^-s + 18.9·17^-s − 76.7i·19^-s + 190.·23^-s + 82.7·25^-s + 138. i·29^-s + 265.·31^-s + 104. i·35^-s + 14.9i·37^-s − 202.·41^-s + 133. i·43^-s − 130.·47^-s + ⋯

L(s) = 1

− 0.581i·5^-s − 0.867·7^-s − 0.765i·11^-s − 0.758i·13^-s + 0.269·17^-s − 0.926i·19^-s + 1.72·23^-s + 0.661·25^-s + 0.884i·29^-s + 1.53·31^-s + 0.504i·35^-s + 0.0665i·37^-s − 0.771·41^-s + 0.473i·43^-s − 0.403·47^-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:	$\chi_{1728} (865, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ -0.707 + 0.707i)$

Particular Values

$L(2)$	$\approx$	$1.482211012$
$L(\frac12)$	$\approx$	$1.482211012$
$L(\frac{5}{2})$		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 + 6.50iT - 125T^{2} $
	7	$ 1 + 16.0T + 343T^{2} $
	11	$ 1 + 27.9iT - 1.33e3T^{2} $
	13	$ 1 + 35.5iT - 2.19e3T^{2} $
	17	$ 1 - 18.9T + 4.91e3T^{2} $
	19	$ 1 + 76.7iT - 6.85e3T^{2} $
	23	$ 1 - 190.T + 1.21e4T^{2} $
	29	$ 1 - 138. iT - 2.43e4T^{2} $
	31	$ 1 - 265.T + 2.97e4T^{2} $
	37	$ 1 - 14.9iT - 5.06e4T^{2} $
	41	$ 1 + 202.T + 6.89e4T^{2} $
	43	$ 1 - 133. iT - 7.95e4T^{2} $
	47	$ 1 + 130.T + 1.03e5T^{2} $
	53	$ 1 - 131. iT - 1.48e5T^{2} $
	59	$ 1 + 477. iT - 2.05e5T^{2} $
	61	$ 1 + 582. iT - 2.26e5T^{2} $
	67	$ 1 - 20iT - 3.00e5T^{2} $
	71	$ 1 - 60.1T + 3.57e5T^{2} $
	73	$ 1 + 321.T + 3.89e5T^{2} $
	79	$ 1 - 210.T + 4.93e5T^{2} $
	83	$ 1 + 587. iT - 5.71e5T^{2} $
	89	$ 1 + 55.5T + 7.04e5T^{2} $
	97	$ 1 + 75.9T + 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

−8.738412191889541185265440336343, −8.018165215903460664009100345178, −6.92558142797995719631260455837, −6.35265938705777251820715839550, −5.26933187460719874271578158123, −4.72495474500457091405350696807, −3.27971440419066121235712220382, −2.90749452878888656753084889767, −1.17482655371208842310231851303, −0.37248752750392918653529003846, 1.15328432784931362057166838428, 2.44887448492930588323585939525, 3.25155509657697070110463879198, 4.22851516049388415065182711884, 5.16065084981600396258619869758, 6.30340750697476830300737258606, 6.78524615959232163662250190026, 7.50813553393906407494732114739, 8.533946245417675738389301455344, 9.358417148766172055711016768950

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.482211012\)
\(L(\frac12)\)	\(\approx\)	\(1.482211012\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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