LMFDB - L-function 2-12e3-8.5-c3-0-44

Dirichlet series

L(s) = 1

+ 12.4i·5^-s − 12.1·7^-s − 21.6i·11^-s + 15.5i·13^-s + 64.8·17^-s − 49i·19^-s − 62.4·23^-s − 31·25^-s − 24.9i·29^-s − 24.2·31^-s − 151. i·35^-s + 102. i·37^-s + 346.·41^-s − 260i·43^-s − 362.·47^-s + ⋯

L(s) = 1

+ 1.11i·5^-s − 0.654·7^-s − 0.592i·11^-s + 0.332i·13^-s + 0.925·17^-s − 0.591i·19^-s − 0.566·23^-s − 0.247·25^-s − 0.159i·29^-s − 0.140·31^-s − 0.731i·35^-s + 0.454i·37^-s + 1.31·41^-s − 0.922i·43^-s − 1.12·47^-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:	$\chi_{1728} (865, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ 0.965 - 0.258i)$

Particular Values

$L(2)$	$\approx$	$1.775790857$
$L(\frac12)$	$\approx$	$1.775790857$
$L(\frac{5}{2})$		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 - 12.4iT - 125T^{2} $
	7	$ 1 + 12.1T + 343T^{2} $
	11	$ 1 + 21.6iT - 1.33e3T^{2} $
	13	$ 1 - 15.5iT - 2.19e3T^{2} $
	17	$ 1 - 64.8T + 4.91e3T^{2} $
	19	$ 1 + 49iT - 6.85e3T^{2} $
	23	$ 1 + 62.4T + 1.21e4T^{2} $
	29	$ 1 + 24.9iT - 2.43e4T^{2} $
	31	$ 1 + 24.2T + 2.97e4T^{2} $
	37	$ 1 - 102. iT - 5.06e4T^{2} $
	41	$ 1 - 346.T + 6.89e4T^{2} $
	43	$ 1 + 260iT - 7.95e4T^{2} $
	47	$ 1 + 362.T + 1.03e5T^{2} $
	53	$ 1 + 574. iT - 1.48e5T^{2} $
	59	$ 1 + 324. iT - 2.05e5T^{2} $
	61	$ 1 - 174. iT - 2.26e5T^{2} $
	67	$ 1 - 241iT - 3.00e5T^{2} $
	71	$ 1 - 249.T + 3.57e5T^{2} $
	73	$ 1 - 353T + 3.89e5T^{2} $
	79	$ 1 - 5.19T + 4.93e5T^{2} $
	83	$ 1 + 1.03e3iT - 5.71e5T^{2} $
	89	$ 1 - 800.T + 7.04e5T^{2} $
	97	$ 1 - 1.11e3T + 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

−9.078463724054519897306402518119, −8.121213866948600409876653214017, −7.31717379470243174485509232727, −6.54798341008605763165115893380, −6.00483743906019218580650511684, −4.93618964324664123223409298246, −3.65719501559296956532508313069, −3.13061890625864066466978713679, −2.11440620887095422460155704881, −0.58893958499190445932714812555, 0.67936230660415850343747577523, 1.67383773945118815475685424529, 2.97161738130859140239068881109, 3.97355885281839026944233920296, 4.83243023935587112409384236030, 5.65740770477790141410135332474, 6.39957696558536390964093724207, 7.55136783942266277871605814772, 8.072724861889323183386795929031, 9.040604338143270155835454084961

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.775790857\)
\(L(\frac12)\)	\(\approx\)	\(1.775790857\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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