LMFDB - L-function 2-2e5-8.5-c17-0-10

Dirichlet series

L(s) = 1

− 1.37e4i·3^-s + 9.63e4i·5^-s + 1.47e7·7^-s − 6.09e7·9^-s − 6.68e8i·11^-s − 1.72e8i·13^-s + 1.32e9·15^-s + 1.92e10·17^-s + 1.11e11i·19^-s − 2.03e11i·21^-s + 5.92e11·23^-s + 7.53e11·25^-s − 9.40e11i·27^-s + 2.95e12i·29^-s + 7.53e12·31^-s + ⋯

L(s) = 1

− 1.21i·3^-s + 0.110i·5^-s + 0.968·7^-s − 0.471·9^-s − 0.940i·11^-s − 0.0586i·13^-s + 0.133·15^-s + 0.670·17^-s + 1.50i·19^-s − 1.17i·21^-s + 1.57·23^-s + 0.987·25^-s − 0.640i·27^-s + 1.09i·29^-s + 1.58·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0506 + 0.998i)\, \overline{\Lambda}(18-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.0506 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$32$ = $2^{5}$
Sign:	$0.0506 + 0.998i$
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:	$(2,\ 32,\ (\ :17/2),\ 0.0506 + 0.998i)$

Particular Values

$L(9)$	$\approx$	$2.659979793$
$L(\frac12)$	$\approx$	$2.659979793$
$L(\frac{19}{2})$		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 + 1.37e4iT - 1.29e8T^{2} $
	5	$ 1 - 9.63e4iT - 7.62e11T^{2} $
	7	$ 1 - 1.47e7T + 2.32e14T^{2} $
	11	$ 1 + 6.68e8iT - 5.05e17T^{2} $
	13	$ 1 + 1.72e8iT - 8.65e18T^{2} $
	17	$ 1 - 1.92e10T + 8.27e20T^{2} $
	19	$ 1 - 1.11e11iT - 5.48e21T^{2} $
	23	$ 1 - 5.92e11T + 1.41e23T^{2} $
	29	$ 1 - 2.95e12iT - 7.25e24T^{2} $
	31	$ 1 - 7.53e12T + 2.25e25T^{2} $
	37	$ 1 + 2.20e13iT - 4.56e26T^{2} $
	41	$ 1 + 8.98e13T + 2.61e27T^{2} $
	43	$ 1 - 1.45e13iT - 5.87e27T^{2} $
	47	$ 1 - 7.43e13T + 2.66e28T^{2} $
	53	$ 1 + 2.84e14iT - 2.05e29T^{2} $
	59	$ 1 + 1.09e15iT - 1.27e30T^{2} $
	61	$ 1 + 4.14e14iT - 2.24e30T^{2} $
	67	$ 1 + 2.20e15iT - 1.10e31T^{2} $
	71	$ 1 + 6.67e15T + 2.96e31T^{2} $
	73	$ 1 + 1.64e15T + 4.74e31T^{2} $
	79	$ 1 - 1.46e16T + 1.81e32T^{2} $
	83	$ 1 + 3.05e16iT - 4.21e32T^{2} $
	89	$ 1 - 4.48e16T + 1.37e33T^{2} $
	97	$ 1 + 3.26e16T + 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

−12.73162836330234832677506449288, −11.72396921620013065843210127661, −10.51929457790959155204954353524, −8.590338258644272283417686879482, −7.69607012823391113391204388441, −6.48553105809164121773647588183, −5.11880703913834097198055968874, −3.20401550254543090069827540601, −1.64466907945476319194433412008, −0.838078457888270047434789602871, 1.13076274457385047178990265244, 2.84831739890395205150000215633, 4.51870732982265324302432568819, 5.00385963371724320497248343098, 7.04013130702226315515753463284, 8.576289125192524199403895248605, 9.723919593976568912427590365578, 10.75719979470849314987528382103, 11.87925414713142853285356141317, 13.45320828189607315889765929812

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 $\F_{q}$

\(L(9)\)	\(\approx\)	\(2.659979793\)
\(L(\frac12)\)	\(\approx\)	\(2.659979793\)
\(L(\frac{19}{2})\)		not available
\(L(1)\)		not available

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