LMFDB - L-function 2-2e5-8.5-c17-0-11

Dirichlet series

L(s) = 1

− 4.83e3i·3^-s + 5.24e5i·5^-s − 1.57e7·7^-s + 1.05e8·9^-s + 2.44e8i·11^-s + 2.66e9i·13^-s + 2.53e9·15^-s − 2.82e10·17^-s − 7.79e10i·19^-s + 7.61e10i·21^-s + 3.66e11·23^-s + 4.87e11·25^-s − 1.13e12i·27^-s − 2.78e11i·29^-s − 3.68e12·31^-s + ⋯

L(s) = 1

− 0.425i·3^-s + 0.600i·5^-s − 1.03·7^-s + 0.819·9^-s + 0.344i·11^-s + 0.905i·13^-s + 0.255·15^-s − 0.983·17^-s − 1.05i·19^-s + 0.439i·21^-s + 0.976·23^-s + 0.638·25^-s − 0.773i·27^-s − 0.103i·29^-s − 0.776·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.702i)\, \overline{\Lambda}(18-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.712 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$32$ = $2^{5}$
Sign:	$-0.712 + 0.702i$
Analytic conductor:	$58.6310$
Root analytic conductor:	$7.65709$
Motivic weight:	$17$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{32} (17, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 32,\ (\ :17/2),\ -0.712 + 0.702i)$

Particular Values

$L(9)$	$\approx$	$0.6019392055$
$L(\frac12)$	$\approx$	$0.6019392055$
$L(\frac{19}{2})$		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 $
good	3	$ 1 + 4.83e3iT - 1.29e8T^{2} $
	5	$ 1 - 5.24e5iT - 7.62e11T^{2} $
	7	$ 1 + 1.57e7T + 2.32e14T^{2} $
	11	$ 1 - 2.44e8iT - 5.05e17T^{2} $
	13	$ 1 - 2.66e9iT - 8.65e18T^{2} $
	17	$ 1 + 2.82e10T + 8.27e20T^{2} $
	19	$ 1 + 7.79e10iT - 5.48e21T^{2} $
	23	$ 1 - 3.66e11T + 1.41e23T^{2} $
	29	$ 1 + 2.78e11iT - 7.25e24T^{2} $
	31	$ 1 + 3.68e12T + 2.25e25T^{2} $
	37	$ 1 - 3.50e13iT - 4.56e26T^{2} $
	41	$ 1 + 3.67e13T + 2.61e27T^{2} $
	43	$ 1 + 1.25e14iT - 5.87e27T^{2} $
	47	$ 1 + 1.02e14T + 2.66e28T^{2} $
	53	$ 1 + 4.67e14iT - 2.05e29T^{2} $
	59	$ 1 - 4.08e13iT - 1.27e30T^{2} $
	61	$ 1 + 2.55e15iT - 2.24e30T^{2} $
	67	$ 1 + 2.46e15iT - 1.10e31T^{2} $
	71	$ 1 + 1.09e15T + 2.96e31T^{2} $
	73	$ 1 + 1.21e16T + 4.74e31T^{2} $
	79	$ 1 + 2.49e16T + 1.81e32T^{2} $
	83	$ 1 + 3.68e16iT - 4.21e32T^{2} $
	89	$ 1 + 4.15e16T + 1.37e33T^{2} $
	97	$ 1 - 7.13e15T + 5.95e33T^{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

−12.79261126139601212121217851379, −11.39086862984795614475755217880, −10.08934403725014572362181838677, −8.952677062756789957407524475196, −6.97428878584095704799859381797, −6.69277404533978973610513831721, −4.63046303239334622084174044881, −3.12895084922730680338730165847, −1.80208167094820242193272112427, −0.15876104805499754985914592263, 1.21618687861713950882223838105, 3.06324138531732843180936771285, 4.30954352543747225563891667532, 5.69472021140097195358491367957, 7.12790111180927356490220635972, 8.722150020762228763737975984150, 9.782044814526551127769678977721, 10.86097374923664276153341371261, 12.64841371240923958604792846474, 13.12448447064494245371630462265

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 $\F_{q}$

\(L(9)\)	\(\approx\)	\(0.6019392055\)
\(L(\frac12)\)	\(\approx\)	\(0.6019392055\)
\(L(\frac{19}{2})\)		not available
\(L(1)\)		not available

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