LMFDB - L-function 2-3060-17.16-c1-0-26

Dirichlet series

L(s) = 1

− i·5^-s − 3.82i·7^-s − 5.82i·11^-s + 4.82·13^-s + (−3 − 2.82i)17^-s + 3·19^-s + 8.48i·23^-s − 25^-s − 4.17i·29^-s − 6.82i·31^-s − 3.82·35^-s − 9.48i·37^-s + 5.82i·41^-s + 8·43^-s + 8.65·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s − 1.44i·7^-s − 1.75i·11^-s + 1.33·13^-s + (−0.727 − 0.685i)17^-s + 0.688·19^-s + 1.76i·23^-s − 0.200·25^-s − 0.774i·29^-s − 1.22i·31^-s − 0.647·35^-s − 1.55i·37^-s + 0.910i·41^-s + 1.21·43^-s + 1.26·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3060$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 17$
Sign:	$-0.685 + 0.727i$
Analytic conductor:	$24.4342$
Root analytic conductor:	$4.94309$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3060} (1801, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3060,\ (\ :1/2),\ -0.685 + 0.727i)$

Particular Values

$L(1)$	$\approx$	$1.727173977$
$L(\frac12)$	$\approx$	$1.727173977$
$L(\frac{3}{2})$		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 $
	5	$ 1 + iT $
	17	$ 1 + (3 + 2.82i)T $
good	7	$ 1 + 3.82iT - 7T^{2} $
	11	$ 1 + 5.82iT - 11T^{2} $
	13	$ 1 - 4.82T + 13T^{2} $
	19	$ 1 - 3T + 19T^{2} $
	23	$ 1 - 8.48iT - 23T^{2} $
	29	$ 1 + 4.17iT - 29T^{2} $
	31	$ 1 + 6.82iT - 31T^{2} $
	37	$ 1 + 9.48iT - 37T^{2} $
	41	$ 1 - 5.82iT - 41T^{2} $
	43	$ 1 - 8T + 43T^{2} $
	47	$ 1 - 8.65T + 47T^{2} $
	53	$ 1 + 8.31T + 53T^{2} $
	59	$ 1 + 5.17T + 59T^{2} $
	61	$ 1 - 8.48iT - 61T^{2} $
	67	$ 1 + 2.48T + 67T^{2} $
	71	$ 1 - 9.65iT - 71T^{2} $
	73	$ 1 - 15.8iT - 73T^{2} $
	79	$ 1 - 2iT - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 + 14.1T + 89T^{2} $
	97	$ 1 + 14iT - 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

−8.396746772966987140865837311421, −7.68448136287558187134530352603, −7.08228931168322321943550436702, −5.89838778382846321458175610686, −5.66575243670484770676074175496, −4.22133671786689407912540976606, −3.85306673844821763004654845030, −2.90115356044360749449942373833, −1.27939232948021619726252348062, −0.58819544708828148594318039761, 1.59741315811936641264373071003, 2.39637780606809569600615847040, 3.29753916105970420178875216626, 4.43187130570860738349396921904, 5.09296862074861023685278723820, 6.16670320250255873035098356007, 6.54966847258376441831059510465, 7.45889993787581502664720149714, 8.424276377410773078953906973490, 8.914116323207520047396647470538

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.727173977\)
\(L(\frac12)\)	\(\approx\)	\(1.727173977\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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