LMFDB - L-function 1-1560-1560.467-r1-0-0

Dirichlet series

L(s) = 1

+ i·7^-s + 11^-s − i·17^-s + 19^-s − i·23^-s − 29^-s + 31^-s + i·37^-s + 41^-s − i·43^-s − i·47^-s − 49^-s − i·53^-s − 59^-s − 61^-s + ⋯

L(s) = 1

+ i·7^-s + 11^-s − i·17^-s + 19^-s − i·23^-s − 29^-s + 31^-s + i·37^-s + 41^-s − i·43^-s − i·47^-s − 49^-s − i·53^-s − 59^-s − 61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$1$
Conductor:	$1560$ = $2^{3} \cdot 3 \cdot 5 \cdot 13$
Sign:	$0.525 - 0.850i$
Analytic conductor:	$167.645$
Root analytic conductor:	$167.645$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1560} (467, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1560,\ (1:\ ),\ 0.525 - 0.850i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.799170521 - 1.003103263i$
$L(\frac12)$	$\approx$	$1.799170521 - 1.003103263i$
$L(1)$	$\approx$	$1.146467649 - 0.03948647112i$
$L(1)$	$\approx$	$1.146467649 - 0.03948647112i$

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	5	$ 1 $
	13	$ 1 $
good	7	$ 1 $
	11	$ 1 $
	17	$ 1 $
	19	$ 1 + iT $
	23	$ 1 $
	29	$ 1 $
	31	$ 1 $
	37	$ 1 + T $
	41	$ 1 $
	43	$ 1 $
	47	$ 1 $
	53	$ 1 $
	59	$ 1 $
	61	$ 1 - iT $
	67	$ 1 $
	71	$ 1 + T $
	73	$ 1 $
	79	$ 1 $
	83	$ 1 $
	89	$ 1 - iT $
	97	$ 1 $
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$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−20.41907946027901716374835045525, −19.56643999915600105624013534855, −19.33055687144324215176058255063, −18.061578969264000127082674607087, −17.42199306902543269236360061284, −16.82564440000558680553608372973, −16.07810473323563252954445799561, −15.169609725665307328205838373927, −14.34066598384054307097369502219, −13.77551315184970824691124183707, −12.97106674778104895099296034877, −12.10567147510939449377021826266, −11.25899067643252340650677667322, −10.63999846134398940275450386887, −9.6293276212519853657109715321, −9.14099644194779939029256251851, −7.872707662840838738053675597447, −7.41900263038927182809075474696, −6.40780948831327924697139081564, −5.70607948687414414324572233726, −4.48039558111455361497285166350, −3.87479071477491748040678078911, −3.02598382792242171427026739057, −1.59717652533431930257069609779, −0.98498519995550861171798127449, 0.43132277722551505253756874066, 1.56769870847794463996613186162, 2.5839137580230249067698359071, 3.38032348892482090496928842363, 4.50296425071810117306561286337, 5.29761365847361060811209536886, 6.16752663112759837579754120409, 6.92543929139462119061489452973, 7.87900516412053445813173300172, 8.83877461777573086426777830809, 9.34674891402799509385517813746, 10.16070593287825908492955978138, 11.346968985207855875410919979272, 11.84237353493032324353483364715, 12.47521899807203918371651921518, 13.53428900260737036072828214309, 14.22138296034857244667008308158, 14.9970668286655510370691472615, 15.70330329469705644406201128439, 16.48310798339557124283354287986, 17.1995615977955306263617902891, 18.21178671219624563672294278081, 18.59016119217271350001096653625, 19.45546387359989444201899782187, 20.278752951033736843837150081127

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.799170521 - 1.003103263i\)
\(L(\frac12)\)	\(\approx\)	\(1.799170521 - 1.003103263i\)
\(L(1)\)	\(\approx\)	\(1.146467649 - 0.03948647112i\)
\(L(1)\)	\(\approx\)	\(1.146467649 - 0.03948647112i\)

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