LMFDB - L-function 2-5472-76.75-c1-0-24

Dirichlet series

L(s) = 1

− 1.27·5^-s − 2.45i·7^-s + 4.60i·11^-s − 5.38i·13^-s − 2.30·17^-s + (0.0756 + 4.35i)19^-s − 3.46i·23^-s − 3.38·25^-s + 4.46i·29^-s + 0.666·31^-s + 3.11i·35^-s − 4.23i·37^-s + 7.19i·41^-s + 1.91i·43^-s − 1.97i·47^-s + ⋯

L(s) = 1

− 0.568·5^-s − 0.927i·7^-s + 1.38i·11^-s − 1.49i·13^-s − 0.559·17^-s + (0.0173 + 0.999i)19^-s − 0.723i·23^-s − 0.677·25^-s + 0.828i·29^-s + 0.119·31^-s + 0.526i·35^-s − 0.696i·37^-s + 1.12i·41^-s + 0.291i·43^-s − 0.288i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5472$ = $2^{5} \cdot 3^{2} \cdot 19$
Sign:	$0.694 - 0.719i$
Analytic conductor:	$43.6941$
Root analytic conductor:	$6.61015$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5472} (2431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5472,\ (\ :1/2),\ 0.694 - 0.719i)$

Particular Values

$L(1)$	$\approx$	$1.207387070$
$L(\frac12)$	$\approx$	$1.207387070$
$L(\frac{3}{2})$		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 $
	19	$ 1 + (-0.0756 - 4.35i)T $
good	5	$ 1 + 1.27T + 5T^{2} $
	7	$ 1 + 2.45iT - 7T^{2} $
	11	$ 1 - 4.60iT - 11T^{2} $
	13	$ 1 + 5.38iT - 13T^{2} $
	17	$ 1 + 2.30T + 17T^{2} $
	23	$ 1 + 3.46iT - 23T^{2} $
	29	$ 1 - 4.46iT - 29T^{2} $
	31	$ 1 - 0.666T + 31T^{2} $
	37	$ 1 + 4.23iT - 37T^{2} $
	41	$ 1 - 7.19iT - 41T^{2} $
	43	$ 1 - 1.91iT - 43T^{2} $
	47	$ 1 + 1.97iT - 47T^{2} $
	53	$ 1 - 7.96iT - 53T^{2} $
	59	$ 1 - 0.672T + 59T^{2} $
	61	$ 1 - 6.33T + 61T^{2} $
	67	$ 1 - 3.25T + 67T^{2} $
	71	$ 1 - 5.48T + 71T^{2} $
	73	$ 1 + 5.78T + 73T^{2} $
	79	$ 1 + 6.85T + 79T^{2} $
	83	$ 1 + 7.51iT - 83T^{2} $
	89	$ 1 + 1.80iT - 89T^{2} $
	97	$ 1 - 6.69iT - 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

−8.003958263961502322319103654683, −7.60819648509174571975932501208, −7.01170920586187969495295358434, −6.16969211024926857334128672637, −5.27003009345614512625940377874, −4.45782957626266632122380506929, −3.91696377996435752892706527925, −3.03714627171715803460314517652, −1.96779662011267177072731497049, −0.816839570255398077349584000827, 0.41498259916444562158202899009, 1.86513649048338172715774803361, 2.68219335576624715379121594834, 3.64231352499243090177457996256, 4.30094708828081771954426162570, 5.23430692211653754303665072602, 5.93417608309665740077932765442, 6.63212930704565863909836111931, 7.31244860651255748707059464100, 8.275425317962750323575184301377

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.207387070\)
\(L(\frac12)\)	\(\approx\)	\(1.207387070\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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