LMFDB - L-function 2-56e2-28.27-c1-0-44

Dirichlet series

L(s) = 1

− 2.14·3^-s − 2.61i·5^-s + 1.58·9^-s + 3.95i·11^-s − 1.08i·13^-s + 5.59i·15^-s + 0.317i·17^-s − 5.16·19^-s − 2.31i·23^-s − 1.82·25^-s + 3.02·27^-s + 6.82·29^-s + 6.05·31^-s − 8.47i·33^-s + 4·37^-s + ⋯

L(s) = 1

− 1.23·3^-s − 1.16i·5^-s + 0.528·9^-s + 1.19i·11^-s − 0.300i·13^-s + 1.44i·15^-s + 0.0768i·17^-s − 1.18·19^-s − 0.483i·23^-s − 0.365·25^-s + 0.582·27^-s + 1.26·29^-s + 1.08·31^-s − 1.47i·33^-s + 0.657·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3136$ = $2^{6} \cdot 7^{2}$
Sign:	$-0.156 + 0.987i$
Analytic conductor:	$25.0410$
Root analytic conductor:	$5.00410$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3136} (3135, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3136,\ (\ :1/2),\ -0.156 + 0.987i)$

Particular Values

$L(1)$	$\approx$	$0.7975467068$
$L(\frac12)$	$\approx$	$0.7975467068$
$L(\frac{3}{2})$		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 + 2.14T + 3T^{2} $
	5	$ 1 + 2.61iT - 5T^{2} $
	11	$ 1 - 3.95iT - 11T^{2} $
	13	$ 1 + 1.08iT - 13T^{2} $
	17	$ 1 - 0.317iT - 17T^{2} $
	19	$ 1 + 5.16T + 19T^{2} $
	23	$ 1 + 2.31iT - 23T^{2} $
	29	$ 1 - 6.82T + 29T^{2} $
	31	$ 1 - 6.05T + 31T^{2} $
	37	$ 1 - 4T + 37T^{2} $
	41	$ 1 + 2.29iT - 41T^{2} $
	43	$ 1 - 7.23iT - 43T^{2} $
	47	$ 1 + 4.28T + 47T^{2} $
	53	$ 1 + 10.4T + 53T^{2} $
	59	$ 1 - 11.2T + 59T^{2} $
	61	$ 1 + 5.41iT - 61T^{2} $
	67	$ 1 + 3.27iT - 67T^{2} $
	71	$ 1 - 71T^{2} $
	73	$ 1 + 14.0iT - 73T^{2} $
	79	$ 1 - 7.91iT - 79T^{2} $
	83	$ 1 - 9.45T + 83T^{2} $
	89	$ 1 - 5.99iT - 89T^{2} $
	97	$ 1 - 9.23iT - 97T^{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

−8.399441594770563439196905996326, −7.87219638927341898079714179119, −6.56435809170821918012713286746, −6.37662379514331281768635068375, −5.20852382606521577160080026680, −4.76499002147272092024345064171, −4.22193232445292282119348081359, −2.65492298071830704570442344794, −1.43439989388875242403955704948, −0.39566879361475298954293762748, 0.901712979031246792847353060272, 2.46150210732057981425940235589, 3.27331356086466243900511380116, 4.31591534536871193478291185631, 5.20392482791887495075799017162, 6.14414697696848352704688863670, 6.39920358163851396473427490643, 7.08821254064153995476984084848, 8.171437521172317144693689467111, 8.762304448504709644810281850125

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7975467068\)
\(L(\frac12)\)	\(\approx\)	\(0.7975467068\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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