LMFDB - L-function 2-28e2-7.6-c4-0-18

Dirichlet series

L(s) = 1

− 16.1i·3^-s − 7.83i·5^-s − 179.·9^-s − 30.5·11^-s + 162. i·13^-s − 126.·15^-s + 345. i·17^-s − 263. i·19^-s − 213.·23^-s + 563.·25^-s + 1.58e3i·27^-s − 1.01e3·29^-s + 273. i·31^-s + 492. i·33^-s + 1.54e3·37^-s + ⋯

L(s) = 1

− 1.79i·3^-s − 0.313i·5^-s − 2.21·9^-s − 0.252·11^-s + 0.960i·13^-s − 0.561·15^-s + 1.19i·17^-s − 0.731i·19^-s − 0.402·23^-s + 0.901·25^-s + 2.17i·27^-s − 1.20·29^-s + 0.284i·31^-s + 0.452i·33^-s + 1.12·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(5-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$0.912 + 0.409i$
Analytic conductor:	$81.0420$
Root analytic conductor:	$9.00233$
Motivic weight:	$4$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{784} (97, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 784,\ (\ :2),\ 0.912 + 0.409i)$

Particular Values

$L(\frac{5}{2})$	$\approx$	$1.423021433$
$L(\frac12)$	$\approx$	$1.423021433$
$L(3)$		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 + 16.1iT - 81T^{2} $
	5	$ 1 + 7.83iT - 625T^{2} $
	11	$ 1 + 30.5T + 1.46e4T^{2} $
	13	$ 1 - 162. iT - 2.85e4T^{2} $
	17	$ 1 - 345. iT - 8.35e4T^{2} $
	19	$ 1 + 263. iT - 1.30e5T^{2} $
	23	$ 1 + 213.T + 2.79e5T^{2} $
	29	$ 1 + 1.01e3T + 7.07e5T^{2} $
	31	$ 1 - 273. iT - 9.23e5T^{2} $
	37	$ 1 - 1.54e3T + 1.87e6T^{2} $
	41	$ 1 - 531. iT - 2.82e6T^{2} $
	43	$ 1 - 1.47e3T + 3.41e6T^{2} $
	47	$ 1 - 2.38e3iT - 4.87e6T^{2} $
	53	$ 1 - 3.20e3T + 7.89e6T^{2} $
	59	$ 1 + 3.11e3iT - 1.21e7T^{2} $
	61	$ 1 - 3.94e3iT - 1.38e7T^{2} $
	67	$ 1 - 6.91e3T + 2.01e7T^{2} $
	71	$ 1 + 8.58e3T + 2.54e7T^{2} $
	73	$ 1 - 359. iT - 2.83e7T^{2} $
	79	$ 1 + 2.08e3T + 3.89e7T^{2} $
	83	$ 1 - 1.05e4iT - 4.74e7T^{2} $
	89	$ 1 + 1.04e4iT - 6.27e7T^{2} $
	97	$ 1 - 1.40e3iT - 8.85e7T^{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

−9.373955148307853725212957010119, −8.605027676357942128723137217874, −7.85365919337899629220781277324, −7.06001356203688608844313914138, −6.34234517928668154496120705894, −5.52064913080887141691461188886, −4.18620670568160507291277335068, −2.70777318760967853082535927348, −1.79169584790346343064145247780, −0.883052000407748912940115904081, 0.40137172429730618003178017692, 2.55930230175888973018217738306, 3.40549070194627006073851302713, 4.28890326808742996637862117900, 5.27147168230291258053823334884, 5.84785118022162715458251403429, 7.29294177642518252836693031049, 8.271213444244360439938508707900, 9.180781027257369157010104119010, 9.859252917160547832339234375860

Overview	Random
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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

\(L(\frac{5}{2})\)	\(\approx\)	\(1.423021433\)
\(L(\frac12)\)	\(\approx\)	\(1.423021433\)
\(L(3)\)		not available
\(L(1)\)		not available

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