LMFDB - L-function 2-864-12.11-c1-0-4

Dirichlet series

L(s) = 1

− 0.317i·5^-s + 1.44i·7^-s − 1.09·11^-s + 2.89·13^-s + 3.46i·17^-s + 4.89i·19^-s + 2.82·23^-s + 4.89·25^-s − 9.12i·29^-s + 7.44i·31^-s + 0.460·35^-s + 4.89·37^-s + 9.75i·41^-s + 6.89i·43^-s − 9.12·47^-s + ⋯

L(s) = 1

− 0.142i·5^-s + 0.547i·7^-s − 0.330·11^-s + 0.804·13^-s + 0.840i·17^-s + 1.12i·19^-s + 0.589·23^-s + 0.979·25^-s − 1.69i·29^-s + 1.33i·31^-s + 0.0778·35^-s + 0.805·37^-s + 1.52i·41^-s + 1.05i·43^-s − 1.33·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$6.89907$
Root analytic conductor:	$2.62660$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (863, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :1/2),\ 0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.41333 + 0.585423i$
$L(\frac12)$	$\approx$	$1.41333 + 0.585423i$
$L(\frac{3}{2})$		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 + 0.317iT - 5T^{2} $
	7	$ 1 - 1.44iT - 7T^{2} $
	11	$ 1 + 1.09T + 11T^{2} $
	13	$ 1 - 2.89T + 13T^{2} $
	17	$ 1 - 3.46iT - 17T^{2} $
	19	$ 1 - 4.89iT - 19T^{2} $
	23	$ 1 - 2.82T + 23T^{2} $
	29	$ 1 + 9.12iT - 29T^{2} $
	31	$ 1 - 7.44iT - 31T^{2} $
	37	$ 1 - 4.89T + 37T^{2} $
	41	$ 1 - 9.75iT - 41T^{2} $
	43	$ 1 - 6.89iT - 43T^{2} $
	47	$ 1 + 9.12T + 47T^{2} $
	53	$ 1 + 4.41iT - 53T^{2} $
	59	$ 1 - 9.12T + 59T^{2} $
	61	$ 1 - 4T + 61T^{2} $
	67	$ 1 + 5.10iT - 67T^{2} $
	71	$ 1 - 7.56T + 71T^{2} $
	73	$ 1 - 1.89T + 73T^{2} $
	79	$ 1 + 2iT - 79T^{2} $
	83	$ 1 + 15.8T + 83T^{2} $
	89	$ 1 - 11.9iT - 89T^{2} $
	97	$ 1 + 5T + 97T^{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

−10.23950398020324571363997971030, −9.459504153179297088408505173783, −8.379771792433395125943219967510, −8.061116789994134754054120449869, −6.65889188552986293612440724534, −5.96741432627994098035529935539, −5.00524797058668108928782441205, −3.89232694850032344841025849079, −2.78038271394609597154507654363, −1.38108186880713424715176842510, 0.844880919413339971780223260363, 2.53995173510546861377836244268, 3.60599294638479294984489056865, 4.72862233480786886556007396793, 5.60004235709438400221483706312, 6.87959919537583238305693330098, 7.25923155256101687052158269934, 8.506739071663186118338680525195, 9.132879852509983080242016177830, 10.14112479580990112276185281567

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.41333 + 0.585423i\)
\(L(\frac12)\)	\(\approx\)	\(1.41333 + 0.585423i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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