LMFDB - L-function 2-1224-17.16-c3-0-0

Dirichlet series

L(s) = 1

− 4.89i·5^-s − 4.46i·7^-s − 60.3i·11^-s − 56.1·13^-s + (−25.6 − 65.2i)17^-s − 134.·19^-s − 39.1i·23^-s + 101.·25^-s + 113. i·29^-s + 306. i·31^-s − 21.8·35^-s − 61.9i·37^-s + 317. i·41^-s + 122.·43^-s − 303.·47^-s + ⋯

L(s) = 1

− 0.437i·5^-s − 0.240i·7^-s − 1.65i·11^-s − 1.19·13^-s + (−0.366 − 0.930i)17^-s − 1.62·19^-s − 0.355i·23^-s + 0.808·25^-s + 0.728i·29^-s + 1.77i·31^-s − 0.105·35^-s − 0.275i·37^-s + 1.20i·41^-s + 0.435·43^-s − 0.941·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1224$ = $2^{3} \cdot 3^{2} \cdot 17$
Sign:	$-0.366 - 0.930i$
Analytic conductor:	$72.2183$
Root analytic conductor:	$8.49813$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1224} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1224,\ (\ :3/2),\ -0.366 - 0.930i)$

Particular Values

$L(2)$	$\approx$	$0.1721585519$
$L(\frac12)$	$\approx$	$0.1721585519$
$L(\frac{5}{2})$		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 $
	17	$ 1 + (25.6 + 65.2i)T $
good	5	$ 1 + 4.89iT - 125T^{2} $
	7	$ 1 + 4.46iT - 343T^{2} $
	11	$ 1 + 60.3iT - 1.33e3T^{2} $
	13	$ 1 + 56.1T + 2.19e3T^{2} $
	19	$ 1 + 134.T + 6.85e3T^{2} $
	23	$ 1 + 39.1iT - 1.21e4T^{2} $
	29	$ 1 - 113. iT - 2.43e4T^{2} $
	31	$ 1 - 306. iT - 2.97e4T^{2} $
	37	$ 1 + 61.9iT - 5.06e4T^{2} $
	41	$ 1 - 317. iT - 6.89e4T^{2} $
	43	$ 1 - 122.T + 7.95e4T^{2} $
	47	$ 1 + 303.T + 1.03e5T^{2} $
	53	$ 1 + 133.T + 1.48e5T^{2} $
	59	$ 1 - 130.T + 2.05e5T^{2} $
	61	$ 1 + 772. iT - 2.26e5T^{2} $
	67	$ 1 - 378.T + 3.00e5T^{2} $
	71	$ 1 - 465. iT - 3.57e5T^{2} $
	73	$ 1 - 664. iT - 3.89e5T^{2} $
	79	$ 1 + 925. iT - 4.93e5T^{2} $
	83	$ 1 - 723.T + 5.71e5T^{2} $
	89	$ 1 + 889.T + 7.04e5T^{2} $
	97	$ 1 - 1.50e3iT - 9.12e5T^{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

−9.470286299286958924540143290385, −8.710479782598043420487210751488, −8.191557290386423960435114236461, −7.01933946832812827166687180779, −6.38915093783272875736157531403, −5.21231758803651666784807212285, −4.62720890141179960803230829215, −3.38516340343789750736744610534, −2.45628996849583967475547175198, −0.976775590875215904300141308628, 0.04444631125374599613371189514, 1.97638446479594445419056753272, 2.48460066056287582844577552426, 4.06050452199381977972104530895, 4.63256931686231988651965210189, 5.79805594002326384061032237085, 6.72622061776825786431533190818, 7.35999318393912157033156149571, 8.221263070435392708313259222204, 9.214357569488415816542854080423

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.1721585519\)
\(L(\frac12)\)	\(\approx\)	\(0.1721585519\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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