LMFDB - L-function 2-2720-17.16-c1-0-18

Dirichlet series

L(s) = 1

− 1.67i·3^-s − i·5^-s + 4.87i·7^-s + 0.199·9^-s + 1.43i·11^-s − 3.49·13^-s − 1.67·15^-s + (−2.09 − 3.54i)17^-s + 1.25·19^-s + 8.15·21^-s − 4.27i·23^-s − 25^-s − 5.35i·27^-s − 2.30i·29^-s + 8.16i·31^-s + ⋯

L(s) = 1

− 0.966i·3^-s − 0.447i·5^-s + 1.84i·7^-s + 0.0664·9^-s + 0.432i·11^-s − 0.969·13^-s − 0.432·15^-s + (−0.508 − 0.860i)17^-s + 0.287·19^-s + 1.77·21^-s − 0.891i·23^-s − 0.200·25^-s − 1.03i·27^-s − 0.427i·29^-s + 1.46i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2720$ = $2^{5} \cdot 5 \cdot 17$
Sign:	$0.860 - 0.508i$
Analytic conductor:	$21.7193$
Root analytic conductor:	$4.66039$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2720} (1121, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2720,\ (\ :1/2),\ 0.860 - 0.508i)$

Particular Values

$L(1)$	$\approx$	$1.521134790$
$L(\frac12)$	$\approx$	$1.521134790$
$L(\frac{3}{2})$		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 $
	5	$ 1 + iT $
	17	$ 1 + (2.09 + 3.54i)T $
good	3	$ 1 + 1.67iT - 3T^{2} $
	7	$ 1 - 4.87iT - 7T^{2} $
	11	$ 1 - 1.43iT - 11T^{2} $
	13	$ 1 + 3.49T + 13T^{2} $
	19	$ 1 - 1.25T + 19T^{2} $
	23	$ 1 + 4.27iT - 23T^{2} $
	29	$ 1 + 2.30iT - 29T^{2} $
	31	$ 1 - 8.16iT - 31T^{2} $
	37	$ 1 - 6.75iT - 37T^{2} $
	41	$ 1 - 6.23iT - 41T^{2} $
	43	$ 1 - 10.8T + 43T^{2} $
	47	$ 1 - 5.26T + 47T^{2} $
	53	$ 1 - 5.11T + 53T^{2} $
	59	$ 1 - 7.32T + 59T^{2} $
	61	$ 1 - 14.0iT - 61T^{2} $
	67	$ 1 - 7.18T + 67T^{2} $
	71	$ 1 - 11.1iT - 71T^{2} $
	73	$ 1 - 15.1iT - 73T^{2} $
	79	$ 1 - 10.5iT - 79T^{2} $
	83	$ 1 - 5.75T + 83T^{2} $
	89	$ 1 + 5.87T + 89T^{2} $
	97	$ 1 + 0.696iT - 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

−8.726239211717153102085357312634, −8.266020825847877977734743422603, −7.25380969750267246068543861452, −6.75460539693920083406248349493, −5.80446016427373577786530109168, −5.11454631366646905737299282604, −4.36309992561487518060180600494, −2.58541289573999708890696554377, −2.42457349775544429871396314779, −1.11021306691179762533462726365, 0.55077685393170504070492891252, 2.05122076079071418179833524182, 3.49153805482430086651204440036, 3.92037587468389774328322579119, 4.60316962388728404612100940005, 5.56707596343834528253335153480, 6.57212171618504595171674659135, 7.49651093379324703597680607800, 7.63531206865572717232385553800, 9.029846767806673016725758853179

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

\(L(1)\)	\(\approx\)	\(1.521134790\)
\(L(\frac12)\)	\(\approx\)	\(1.521134790\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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