LMFDB - L-function 2-2640-5.4-c1-0-47

Dirichlet series

L(s) = 1

− i·3^-s + (2.17 − 0.539i)5^-s − 0.290i·7^-s − 9^-s + 11^-s − 6.97i·13^-s + (−0.539 − 2.17i)15^-s + 4.78i·17^-s + 7.75·19^-s − 0.290·21^-s + 4i·23^-s + (4.41 − 2.34i)25^-s + i·27^-s + 7.41·29^-s − 6.34·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + (0.970 − 0.241i)5^-s − 0.109i·7^-s − 0.333·9^-s + 0.301·11^-s − 1.93i·13^-s + (−0.139 − 0.560i)15^-s + 1.16i·17^-s + 1.77·19^-s − 0.0634·21^-s + 0.834i·23^-s + (0.883 − 0.468i)25^-s + 0.192i·27^-s + 1.37·29^-s − 1.13·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2640$ = $2^{4} \cdot 3 \cdot 5 \cdot 11$
Sign:	$0.241 + 0.970i$
Analytic conductor:	$21.0805$
Root analytic conductor:	$4.59135$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2640} (529, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2640,\ (\ :1/2),\ 0.241 + 0.970i)$

Particular Values

$L(1)$	$\approx$	$2.302306177$
$L(\frac12)$	$\approx$	$2.302306177$
$L(\frac{3}{2})$		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 + iT $
	5	$ 1 + (-2.17 + 0.539i)T $
	11	$ 1 - T $
good	7	$ 1 + 0.290iT - 7T^{2} $
	13	$ 1 + 6.97iT - 13T^{2} $
	17	$ 1 - 4.78iT - 17T^{2} $
	19	$ 1 - 7.75T + 19T^{2} $
	23	$ 1 - 4iT - 23T^{2} $
	29	$ 1 - 7.41T + 29T^{2} $
	31	$ 1 + 6.34T + 31T^{2} $
	37	$ 1 + 3.41iT - 37T^{2} $
	41	$ 1 + 7.41T + 41T^{2} $
	43	$ 1 - 0.290iT - 43T^{2} $
	47	$ 1 + 5.26iT - 47T^{2} $
	53	$ 1 + 5.75iT - 53T^{2} $
	59	$ 1 - 3.60T + 59T^{2} $
	61	$ 1 + 6.68T + 61T^{2} $
	67	$ 1 + 6.15iT - 67T^{2} $
	71	$ 1 - 5.07T + 71T^{2} $
	73	$ 1 - 1.12iT - 73T^{2} $
	79	$ 1 + 0.921T + 79T^{2} $
	83	$ 1 - 1.70iT - 83T^{2} $
	89	$ 1 + 4.34T + 89T^{2} $
	97	$ 1 - 4.68iT - 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

−8.612426118961096915249678971397, −7.951557023014428278439246715571, −7.21571676879084708649239357116, −6.32900401284163195807664710552, −5.48813961008071820778141505275, −5.23238595963172697069376454004, −3.65507327557211832922316719412, −2.90823864962864757402690948332, −1.72359574976105642656954148587, −0.841842234192493177452283137314, 1.28855809257500958653215153678, 2.42771267176284252716214710961, 3.25956660427373176245916677486, 4.42885752643919455481606105279, 5.04494629396333221717947691578, 5.90651269742790002367478221782, 6.78019996983690098574598174258, 7.23786660927260251281226627345, 8.572906420334376669082415727836, 9.238236670983916725768442536904

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.302306177\)
\(L(\frac12)\)	\(\approx\)	\(2.302306177\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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