LMFDB - L-function 2-2016-12.11-c1-0-0

Dirichlet series

L(s) = 1

+ 3.62i·5^-s − i·7^-s − 5.03·11^-s + 3.12·13^-s + 6.45i·17^-s − 4i·19^-s − 5.03·23^-s − 8.12·25^-s − 8.65i·29^-s + 10.2i·31^-s + 3.62·35^-s − 2.87·37^-s + 9.27i·41^-s − 4i·43^-s − 1.24·47^-s + ⋯

L(s) = 1

+ 1.62i·5^-s − 0.377i·7^-s − 1.51·11^-s + 0.866·13^-s + 1.56i·17^-s − 0.917i·19^-s − 1.05·23^-s − 1.62·25^-s − 1.60i·29^-s + 1.84i·31^-s + 0.612·35^-s − 0.472·37^-s + 1.44i·41^-s − 0.609i·43^-s − 0.180·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2016$ = $2^{5} \cdot 3^{2} \cdot 7$
Sign:	$-0.985 + 0.169i$
Analytic conductor:	$16.0978$
Root analytic conductor:	$4.01221$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2016} (575, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2016,\ (\ :1/2),\ -0.985 + 0.169i)$

Particular Values

$L(1)$	$\approx$	$0.5154834235$
$L(\frac12)$	$\approx$	$0.5154834235$
$L(\frac{3}{2})$		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 $
	7	$ 1 + iT $
good	5	$ 1 - 3.62iT - 5T^{2} $
	11	$ 1 + 5.03T + 11T^{2} $
	13	$ 1 - 3.12T + 13T^{2} $
	17	$ 1 - 6.45iT - 17T^{2} $
	19	$ 1 + 4iT - 19T^{2} $
	23	$ 1 + 5.03T + 23T^{2} $
	29	$ 1 + 8.65iT - 29T^{2} $
	31	$ 1 - 10.2iT - 31T^{2} $
	37	$ 1 + 2.87T + 37T^{2} $
	41	$ 1 - 9.27iT - 41T^{2} $
	43	$ 1 + 4iT - 43T^{2} $
	47	$ 1 + 1.24T + 47T^{2} $
	53	$ 1 + 8.65iT - 53T^{2} $
	59	$ 1 + 10.0T + 59T^{2} $
	61	$ 1 + 2T + 61T^{2} $
	67	$ 1 - 3.12iT - 67T^{2} $
	71	$ 1 + 9.45T + 71T^{2} $
	73	$ 1 + 7.12T + 73T^{2} $
	79	$ 1 + 13.3iT - 79T^{2} $
	83	$ 1 - 8.83T + 83T^{2} $
	89	$ 1 - 3.62iT - 89T^{2} $
	97	$ 1 + 7.12T + 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

−9.907602847425464083042108670667, −8.547716407209773582025121112526, −7.959439114601936748381724621428, −7.21774778408637939695033695105, −6.37020097609273928087857880704, −5.87559043156913623621638460815, −4.62768859574766001315522889800, −3.58130711526836933841963791498, −2.89808592584442086975616264609, −1.87024855339512328807736430875, 0.17577911668684531039300584873, 1.49158486833138091314604097146, 2.64990487381121945718612767151, 3.88282668834628150728011386364, 4.85162551918703180559624391903, 5.42955175528655444121293461772, 6.05820738288682037858521673912, 7.51421254270460787080087977722, 7.982701617874273988012740694141, 8.801131348606920261411638824872

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5154834235\)
\(L(\frac12)\)	\(\approx\)	\(0.5154834235\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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