LMFDB - L-function 2-28e2-28.27-c5-0-40

Dirichlet series

L(s) = 1

− 7.79·3^-s + 46.3i·5^-s − 182.·9^-s + 756. i·11^-s + 202. i·13^-s − 361. i·15^-s − 1.56e3i·17^-s + 2.40e3·19^-s − 3.40e3i·23^-s + 972.·25^-s + 3.31e3·27^-s + 3.14e3·29^-s + 278.·31^-s − 5.89e3i·33^-s + 7.58e3·37^-s + ⋯

L(s) = 1

− 0.499·3^-s + 0.829i·5^-s − 0.750·9^-s + 1.88i·11^-s + 0.333i·13^-s − 0.414i·15^-s − 1.31i·17^-s + 1.52·19^-s − 1.34i·23^-s + 0.311·25^-s + 0.874·27^-s + 0.694·29^-s + 0.0520·31^-s − 0.942i·33^-s + 0.910·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$0.156 - 0.987i$
Analytic conductor:	$125.740$
Root analytic conductor:	$11.2134$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{784} (783, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 784,\ (\ :5/2),\ 0.156 - 0.987i)$

Particular Values

$L(3)$	$\approx$	$1.667994046$
$L(\frac12)$	$\approx$	$1.667994046$
$L(\frac{7}{2})$		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 $
	7	$ 1 $
good	3	$ 1 + 7.79T + 243T^{2} $
	5	$ 1 - 46.3iT - 3.12e3T^{2} $
	11	$ 1 - 756. iT - 1.61e5T^{2} $
	13	$ 1 - 202. iT - 3.71e5T^{2} $
	17	$ 1 + 1.56e3iT - 1.41e6T^{2} $
	19	$ 1 - 2.40e3T + 2.47e6T^{2} $
	23	$ 1 + 3.40e3iT - 6.43e6T^{2} $
	29	$ 1 - 3.14e3T + 2.05e7T^{2} $
	31	$ 1 - 278.T + 2.86e7T^{2} $
	37	$ 1 - 7.58e3T + 6.93e7T^{2} $
	41	$ 1 - 1.13e4iT - 1.15e8T^{2} $
	43	$ 1 + 4.59e3iT - 1.47e8T^{2} $
	47	$ 1 + 2.34e4T + 2.29e8T^{2} $
	53	$ 1 - 2.75e4T + 4.18e8T^{2} $
	59	$ 1 - 2.56e4T + 7.14e8T^{2} $
	61	$ 1 - 1.58e4iT - 8.44e8T^{2} $
	67	$ 1 + 1.87e4iT - 1.35e9T^{2} $
	71	$ 1 - 233. iT - 1.80e9T^{2} $
	73	$ 1 - 2.65e4iT - 2.07e9T^{2} $
	79	$ 1 + 1.05e5iT - 3.07e9T^{2} $
	83	$ 1 - 7.75e4T + 3.93e9T^{2} $
	89	$ 1 + 2.31e4iT - 5.58e9T^{2} $
	97	$ 1 + 6.02e3iT - 8.58e9T^{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

−9.870168694221620365415424214131, −9.048302768083094077381483622565, −7.80455014367555645843943700105, −6.99798892585676464384795578213, −6.45375726339234839098239640743, −5.16888218422395578089321707208, −4.56751890000933927482503379380, −3.05682328769747956819901145120, −2.33416533972759911982289617824, −0.802237600422404759445521571395, 0.54176649031630004191318245389, 1.22515792120263628627643023663, 2.95474728025406135451976972551, 3.76943713895068503737687556579, 5.27620754419182201543744683442, 5.58865001131421241025477848535, 6.48404774047767447671555850001, 7.912771955096400157824797887152, 8.461075393677262834999302611129, 9.201487978976763261308016519099

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

\(L(3)\)	\(\approx\)	\(1.667994046\)
\(L(\frac12)\)	\(\approx\)	\(1.667994046\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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