LMFDB - L-function 2-24e2-8.5-c3-0-3

Dirichlet series

L(s) = 1

− 18i·11^-s − 90·17^-s + 106i·19^-s + 125·25^-s − 522·41^-s + 290i·43^-s − 343·49^-s + 846i·59^-s − 70i·67^-s − 430·73^-s + 1.35e3i·83^-s − 1.02e3·89^-s − 1.91e3·97^-s + 1.71e3i·107^-s + 270·113^-s + ⋯

L(s) = 1

− 0.493i·11^-s − 1.28·17^-s + 1.27i·19^-s + 25^-s − 1.98·41^-s + 1.02i·43^-s − 49^-s + 1.86i·59^-s − 0.127i·67^-s − 0.689·73^-s + 1.78i·83^-s − 1.22·89^-s − 1.99·97^-s + 1.54i·107^-s + 0.224·113^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$576$ = $2^{6} \cdot 3^{2}$
Sign:	$-0.707 - 0.707i$
Analytic conductor:	$33.9851$
Root analytic conductor:	$5.82967$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{576} (289, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 576,\ (\ :3/2),\ -0.707 - 0.707i)$

Particular Values

$L(2)$	$\approx$	$0.6923090619$
$L(\frac12)$	$\approx$	$0.6923090619$
$L(\frac{5}{2})$		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 $
good	5	$ 1 - 125T^{2} $
	7	$ 1 + 343T^{2} $
	11	$ 1 + 18iT - 1.33e3T^{2} $
	13	$ 1 - 2.19e3T^{2} $
	17	$ 1 + 90T + 4.91e3T^{2} $
	19	$ 1 - 106iT - 6.85e3T^{2} $
	23	$ 1 + 1.21e4T^{2} $
	29	$ 1 - 2.43e4T^{2} $
	31	$ 1 + 2.97e4T^{2} $
	37	$ 1 - 5.06e4T^{2} $
	41	$ 1 + 522T + 6.89e4T^{2} $
	43	$ 1 - 290iT - 7.95e4T^{2} $
	47	$ 1 + 1.03e5T^{2} $
	53	$ 1 - 1.48e5T^{2} $
	59	$ 1 - 846iT - 2.05e5T^{2} $
	61	$ 1 - 2.26e5T^{2} $
	67	$ 1 + 70iT - 3.00e5T^{2} $
	71	$ 1 + 3.57e5T^{2} $
	73	$ 1 + 430T + 3.89e5T^{2} $
	79	$ 1 + 4.93e5T^{2} $
	83	$ 1 - 1.35e3iT - 5.71e5T^{2} $
	89	$ 1 + 1.02e3T + 7.04e5T^{2} $
	97	$ 1 + 1.91e3T + 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

−10.67016050514561331154485571560, −9.819188016137967151442135240238, −8.783472809410815713234616394584, −8.147892752543172415150220471379, −6.97289569395717397899705866122, −6.16734807862017247304645257915, −5.08308260030683094613814473702, −4.00340861063318723668598869248, −2.84269744984927898170892811505, −1.44723192460365610634002241786, 0.20003468293914237381110268328, 1.86993591028199744922661392953, 3.06510397444371321587764166741, 4.42637463352687941659332614256, 5.17684174673876907358549479866, 6.59067541847462047433441894719, 7.08345020592723827539044656372, 8.375186935866125219423757538266, 9.050586010279375456661197467882, 9.990546827521816250209878672378

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.6923090619\)
\(L(\frac12)\)	\(\approx\)	\(0.6923090619\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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