LMFDB - L-function 2-864-24.11-c3-0-33

Dirichlet series

L(s) = 1

− 0.898·5^-s + 20.7i·7^-s + 0.351i·11^-s − 44.6i·13^-s − 58.0i·17^-s + 42.4·19^-s − 196.·23^-s − 124.·25^-s + 71.7·29^-s + 96.1i·31^-s − 18.6i·35^-s + 18.9i·37^-s − 322. i·41^-s + 264.·43^-s + 370.·47^-s + ⋯

L(s) = 1

− 0.0803·5^-s + 1.11i·7^-s + 0.00964i·11^-s − 0.952i·13^-s − 0.828i·17^-s + 0.512·19^-s − 1.78·23^-s − 0.993·25^-s + 0.459·29^-s + 0.557i·31^-s − 0.0898i·35^-s + 0.0840i·37^-s − 1.22i·41^-s + 0.936·43^-s + 1.15·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$0.321 + 0.946i$
Analytic conductor:	$50.9776$
Root analytic conductor:	$7.13986$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :3/2),\ 0.321 + 0.946i)$

Particular Values

$L(2)$	$\approx$	$1.359053549$
$L(\frac12)$	$\approx$	$1.359053549$
$L(\frac{5}{2})$		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 + 0.898T + 125T^{2} $
	7	$ 1 - 20.7iT - 343T^{2} $
	11	$ 1 - 0.351iT - 1.33e3T^{2} $
	13	$ 1 + 44.6iT - 2.19e3T^{2} $
	17	$ 1 + 58.0iT - 4.91e3T^{2} $
	19	$ 1 - 42.4T + 6.85e3T^{2} $
	23	$ 1 + 196.T + 1.21e4T^{2} $
	29	$ 1 - 71.7T + 2.43e4T^{2} $
	31	$ 1 - 96.1iT - 2.97e4T^{2} $
	37	$ 1 - 18.9iT - 5.06e4T^{2} $
	41	$ 1 + 322. iT - 6.89e4T^{2} $
	43	$ 1 - 264.T + 7.95e4T^{2} $
	47	$ 1 - 370.T + 1.03e5T^{2} $
	53	$ 1 - 512.T + 1.48e5T^{2} $
	59	$ 1 + 240. iT - 2.05e5T^{2} $
	61	$ 1 + 526. iT - 2.26e5T^{2} $
	67	$ 1 - 156.T + 3.00e5T^{2} $
	71	$ 1 - 395.T + 3.57e5T^{2} $
	73	$ 1 + 723.T + 3.89e5T^{2} $
	79	$ 1 - 972. iT - 4.93e5T^{2} $
	83	$ 1 + 1.22e3iT - 5.71e5T^{2} $
	89	$ 1 + 1.23e3iT - 7.04e5T^{2} $
	97	$ 1 + 499.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

−9.632314539308490787207849816121, −8.734524775751516515198690145609, −7.991948236858712247280236120648, −7.12537101201123618059277613909, −5.83273786811378152409310626377, −5.46898573470337541311380991436, −4.16339917121316223623037573054, −2.98917209227921931198458990078, −2.04387664460457647478444704037, −0.39532192746389874613670894913, 1.06541069144453351905058578022, 2.29348585359183607113352048380, 3.92758888915887999291362293517, 4.19969169956199168906962530851, 5.67938124140719990455818264266, 6.51366246147306558591736895693, 7.47897907090221238376124674314, 8.087197125370051767008316151144, 9.193456590176175208166424151945, 10.04026964663746109659823440287

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.359053549\)
\(L(\frac12)\)	\(\approx\)	\(1.359053549\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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