LMFDB - L-function 2-864-8.5-c1-0-12

Dirichlet series

L(s) = 1

− 3.36i·5^-s + 2.64·7^-s − 2.16i·11^-s + 4.64i·13^-s + 4.55·17^-s − 6.29i·19^-s − 0.979·23^-s − 6.29·25^-s − 4.33i·29^-s − 2·31^-s − 8.89i·35^-s − 1.35i·37^-s − 11.0·41^-s + 3.29i·43^-s + 10.0·47^-s + ⋯

L(s) = 1

− 1.50i·5^-s + 0.999·7^-s − 0.654i·11^-s + 1.28i·13^-s + 1.10·17^-s − 1.44i·19^-s − 0.204·23^-s − 1.25·25^-s − 0.805i·29^-s − 0.359·31^-s − 1.50i·35^-s − 0.222i·37^-s − 1.72·41^-s + 0.501i·43^-s + 1.47·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$864$ = $2^{5} \cdot 3^{3}$
Sign:	$0.210 + 0.977i$
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:	$(2,\ 864,\ (\ :1/2),\ 0.210 + 0.977i)$

Particular Values

$L(1)$	$\approx$	$1.31483 - 1.06193i$
$L(\frac12)$	$\approx$	$1.31483 - 1.06193i$
$L(\frac{3}{2})$		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 + 3.36iT - 5T^{2} $
	7	$ 1 - 2.64T + 7T^{2} $
	11	$ 1 + 2.16iT - 11T^{2} $
	13	$ 1 - 4.64iT - 13T^{2} $
	17	$ 1 - 4.55T + 17T^{2} $
	19	$ 1 + 6.29iT - 19T^{2} $
	23	$ 1 + 0.979T + 23T^{2} $
	29	$ 1 + 4.33iT - 29T^{2} $
	31	$ 1 + 2T + 31T^{2} $
	37	$ 1 + 1.35iT - 37T^{2} $
	41	$ 1 + 11.0T + 41T^{2} $
	43	$ 1 - 3.29iT - 43T^{2} $
	47	$ 1 - 10.0T + 47T^{2} $
	53	$ 1 + 4.33iT - 53T^{2} $
	59	$ 1 + 11.2iT - 59T^{2} $
	61	$ 1 + 1.93iT - 61T^{2} $
	67	$ 1 - 3iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 11.5T + 73T^{2} $
	79	$ 1 - 8.64T + 79T^{2} $
	83	$ 1 - 2.38iT - 83T^{2} $
	89	$ 1 + 4.55T + 89T^{2} $
	97	$ 1 - 2.29T + 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

−9.748193527528939496558072983622, −9.023453872164857359097694634803, −8.412969660024031181013693112814, −7.66247713079585967482022075006, −6.47161169302262464655364051816, −5.26158941043717062989564040197, −4.80619174370164037973740250967, −3.77925601199069692367879286941, −2.04073969936277866671011076871, −0.892304382562949519357214812331, 1.66147211290975114244345675614, 2.93914083844781911243510315261, 3.80120159264971665645909254916, 5.20421559564216215745206548884, 5.94412322030044062012901462634, 7.12702087147212056734307804349, 7.69033658424448120884289178151, 8.419149101425307613009652244081, 9.866134440698706083308828629917, 10.41114425113324795044799920269

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.31483 - 1.06193i\)
\(L(\frac12)\)	\(\approx\)	\(1.31483 - 1.06193i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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