LMFDB - L-function 2-864-8.5-c1-0-5

Dirichlet series

L(s) = 1

− 0.841i·5^-s − 2.64·7^-s + 3.91i·11^-s − 0.645i·13^-s + 5.59·17^-s + 4.29i·19^-s + 8.66·23^-s + 4.29·25^-s + 7.82i·29^-s − 2·31^-s + 2.22i·35^-s − 6.64i·37^-s + 6.13·41^-s − 7.29i·43^-s + 2.52·47^-s + ⋯

L(s) = 1

− 0.376i·5^-s − 0.999·7^-s + 1.17i·11^-s − 0.179i·13^-s + 1.35·17^-s + 0.984i·19^-s + 1.80·23^-s + 0.858·25^-s + 1.45i·29^-s − 0.359·31^-s + 0.376i·35^-s − 1.09i·37^-s + 0.958·41^-s − 1.11i·43^-s + 0.368·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$0.840 - 0.542i$
Analytic conductor:	$6.89907$
Root analytic conductor:	$2.62660$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{864} (433, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 864,\ (\ :1/2),\ 0.840 - 0.542i)$

Particular Values

$L(1)$	$\approx$	$1.34036 + 0.395156i$
$L(\frac12)$	$\approx$	$1.34036 + 0.395156i$
$L(\frac{3}{2})$		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 + 0.841iT - 5T^{2} $
	7	$ 1 + 2.64T + 7T^{2} $
	11	$ 1 - 3.91iT - 11T^{2} $
	13	$ 1 + 0.645iT - 13T^{2} $
	17	$ 1 - 5.59T + 17T^{2} $
	19	$ 1 - 4.29iT - 19T^{2} $
	23	$ 1 - 8.66T + 23T^{2} $
	29	$ 1 - 7.82iT - 29T^{2} $
	31	$ 1 + 2T + 31T^{2} $
	37	$ 1 + 6.64iT - 37T^{2} $
	41	$ 1 - 6.13T + 41T^{2} $
	43	$ 1 + 7.29iT - 43T^{2} $
	47	$ 1 - 2.52T + 47T^{2} $
	53	$ 1 - 7.82iT - 53T^{2} $
	59	$ 1 + 7.27iT - 59T^{2} $
	61	$ 1 - 13.9iT - 61T^{2} $
	67	$ 1 - 3iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 9.58T + 73T^{2} $
	79	$ 1 - 3.35T + 79T^{2} $
	83	$ 1 - 9.50iT - 83T^{2} $
	89	$ 1 + 5.59T + 89T^{2} $
	97	$ 1 + 8.29T + 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

−10.19743436814023811418707300004, −9.385341204264229788561080949635, −8.764764078746433819968754967572, −7.47050350470450225695036372977, −7.00359176119687560758525480113, −5.78083912106428477652472239609, −5.01524737644631318084769638744, −3.79767811014471942518201921088, −2.83501683812561782257844193409, −1.23169067231171322230580905508, 0.813562671842546427942208295838, 2.87470188038321878037992250783, 3.33511250783158719637926130006, 4.77470403196510118728915199792, 5.87298236588798158724547116792, 6.58580085221747550005916107037, 7.44079749026520585077942212540, 8.483477710440755578225958710882, 9.322053367125596929902245373409, 10.00948123386452870557807616689

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.34036 + 0.395156i\)
\(L(\frac12)\)	\(\approx\)	\(1.34036 + 0.395156i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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