LMFDB - L-function 2-2736-12.11-c1-0-10

Dirichlet series

L(s) = 1

− 1.09i·5^-s + 1.25i·7^-s − 2.92·11^-s − 4.58·13^-s − 0.680i·17^-s − i·19^-s + 6.07·23^-s + 3.80·25^-s − 1.82i·29^-s + 8.58i·31^-s + 1.37·35^-s − 0.510·37^-s − 3.18i·41^-s + 5.38i·43^-s + 10.9·47^-s + ⋯

L(s) = 1

− 0.489i·5^-s + 0.474i·7^-s − 0.881·11^-s − 1.27·13^-s − 0.165i·17^-s − 0.229i·19^-s + 1.26·23^-s + 0.760·25^-s − 0.339i·29^-s + 1.54i·31^-s + 0.232·35^-s − 0.0838·37^-s − 0.498i·41^-s + 0.821i·43^-s + 1.60·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.816 - 0.577i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (2015, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ 0.816 - 0.577i)$

Particular Values

$L(1)$	$\approx$	$1.434509319$
$L(\frac12)$	$\approx$	$1.434509319$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	19	$ 1 + iT $
good	5	$ 1 + 1.09iT - 5T^{2} $
	7	$ 1 - 1.25iT - 7T^{2} $
	11	$ 1 + 2.92T + 11T^{2} $
	13	$ 1 + 4.58T + 13T^{2} $
	17	$ 1 + 0.680iT - 17T^{2} $
	23	$ 1 - 6.07T + 23T^{2} $
	29	$ 1 + 1.82iT - 29T^{2} $
	31	$ 1 - 8.58iT - 31T^{2} $
	37	$ 1 + 0.510T + 37T^{2} $
	41	$ 1 + 3.18iT - 41T^{2} $
	43	$ 1 - 5.38iT - 43T^{2} $
	47	$ 1 - 10.9T + 47T^{2} $
	53	$ 1 - 9.03iT - 53T^{2} $
	59	$ 1 - 3.54T + 59T^{2} $
	61	$ 1 - 4.29T + 61T^{2} $
	67	$ 1 - 11.0iT - 67T^{2} $
	71	$ 1 - 12.1T + 71T^{2} $
	73	$ 1 + 5.25T + 73T^{2} $
	79	$ 1 + 8.26iT - 79T^{2} $
	83	$ 1 - 6.07T + 83T^{2} $
	89	$ 1 - 13.9iT - 89T^{2} $
	97	$ 1 + 7.67T + 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

−8.955502275631259414848909268555, −8.213976863601890409140837890796, −7.32912285389813048119832984054, −6.79985040974515170286357023491, −5.50570974651843119579346898799, −5.13815387260510720728450532753, −4.34474169050336726535085254429, −2.97385554311983678337352179329, −2.39706324109275703007554516488, −0.925805280093667973519141595685, 0.58275200371131209268395488430, 2.20029965995739865112854486463, 2.92306827247157187249490589039, 3.95363714483861917088716247657, 4.92151774625284836013786268225, 5.53585385407350645669083212076, 6.66320535902594234590292770037, 7.26132765868076498758556940207, 7.81612743889353957587018379564, 8.744626324250584918460882212111

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.434509319\)
\(L(\frac12)\)	\(\approx\)	\(1.434509319\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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