LMFDB - L-function 2-3872-8.5-c1-0-98

Dirichlet series

L(s) = 1

− 2.85i·3^-s − 2.60i·5^-s + 0.196·7^-s − 5.16·9^-s + 3.97i·13^-s − 7.45·15^-s + 1.23·17^-s − 5.47i·19^-s − 0.560i·21^-s − 3.19·23^-s − 1.80·25^-s + 6.17i·27^-s + 3.03i·29^-s − 9.83·31^-s − 0.512i·35^-s + ⋯

L(s) = 1

− 1.64i·3^-s − 1.16i·5^-s + 0.0742·7^-s − 1.72·9^-s + 1.10i·13^-s − 1.92·15^-s + 0.300·17^-s − 1.25i·19^-s − 0.122i·21^-s − 0.666·23^-s − 0.360·25^-s + 1.18i·27^-s + 0.563i·29^-s − 1.76·31^-s − 0.0865i·35^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3872$ = $2^{5} \cdot 11^{2}$
Sign:	$0.243 - 0.969i$
Analytic conductor:	$30.9180$
Root analytic conductor:	$5.56040$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3872} (1937, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3872,\ (\ :1/2),\ 0.243 - 0.969i)$

Particular Values

$L(1)$	$\approx$	$0.2928258159$
$L(\frac12)$	$\approx$	$0.2928258159$
$L(\frac{3}{2})$		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 $
	11	$ 1 $
good	3	$ 1 + 2.85iT - 3T^{2} $
	5	$ 1 + 2.60iT - 5T^{2} $
	7	$ 1 - 0.196T + 7T^{2} $
	13	$ 1 - 3.97iT - 13T^{2} $
	17	$ 1 - 1.23T + 17T^{2} $
	19	$ 1 + 5.47iT - 19T^{2} $
	23	$ 1 + 3.19T + 23T^{2} $
	29	$ 1 - 3.03iT - 29T^{2} $
	31	$ 1 + 9.83T + 31T^{2} $
	37	$ 1 - 7.55iT - 37T^{2} $
	41	$ 1 + 7.31T + 41T^{2} $
	43	$ 1 + 3.82iT - 43T^{2} $
	47	$ 1 - 0.507T + 47T^{2} $
	53	$ 1 + 0.0213iT - 53T^{2} $
	59	$ 1 - 5.64iT - 59T^{2} $
	61	$ 1 - 12.9iT - 61T^{2} $
	67	$ 1 - 2.16iT - 67T^{2} $
	71	$ 1 + 4.27T + 71T^{2} $
	73	$ 1 - 2.56T + 73T^{2} $
	79	$ 1 + 5.10T + 79T^{2} $
	83	$ 1 + 2.30iT - 83T^{2} $
	89	$ 1 + 6.22T + 89T^{2} $
	97	$ 1 + 0.832T + 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

−7.85295501670039422854855109277, −7.08670083136240732446836257603, −6.67917004533262325441754559304, −5.70976721869674262319191711194, −5.03446912420936518980417855427, −4.14507277262024451274887054862, −2.89226187924356117245016918531, −1.81522674745073099004026790570, −1.29088061501421870988151453388, −0.082562527925508140720580206290, 2.03519449112753642198895404299, 3.28989818844095098577479407691, 3.46323233814996217893541986383, 4.39051423560053834181576515485, 5.42949252951551065069383224617, 5.80002031214632792367038400613, 6.78955715113315561051978476650, 7.79212205455881247444257322674, 8.273155251224223254086702966536, 9.364864773571584933135748579059

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2928258159\)
\(L(\frac12)\)	\(\approx\)	\(0.2928258159\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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