LMFDB - L-function 2-54e2-3.2-c2-0-44

Dirichlet series

L(s) = 1

− 5.21i·5^-s − 11.6·7^-s − 17.0i·11^-s + 12.9·13^-s + 28.3i·17^-s + 27.8·19^-s − 9.47i·23^-s − 2.21·25^-s + 12.9i·29^-s + 14.9·31^-s + 60.8i·35^-s + 38.6·37^-s + 1.93i·41^-s + 52.6·43^-s + 2.59i·47^-s + ⋯

L(s) = 1

− 1.04i·5^-s − 1.66·7^-s − 1.54i·11^-s + 0.999·13^-s + 1.66i·17^-s + 1.46·19^-s − 0.412i·23^-s − 0.0885·25^-s + 0.447i·29^-s + 0.483·31^-s + 1.73i·35^-s + 1.04·37^-s + 0.0472i·41^-s + 1.22·43^-s + 0.0552i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2916$ = $2^{2} \cdot 3^{6}$
Sign:	$i$
Analytic conductor:	$79.4552$
Root analytic conductor:	$8.91376$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2916} (1457, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2916,\ (\ :1),\ i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.717499038$
$L(\frac12)$	$\approx$	$1.717499038$
$L(2)$		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 + 5.21iT - 25T^{2} $
	7	$ 1 + 11.6T + 49T^{2} $
	11	$ 1 + 17.0iT - 121T^{2} $
	13	$ 1 - 12.9T + 169T^{2} $
	17	$ 1 - 28.3iT - 289T^{2} $
	19	$ 1 - 27.8T + 361T^{2} $
	23	$ 1 + 9.47iT - 529T^{2} $
	29	$ 1 - 12.9iT - 841T^{2} $
	31	$ 1 - 14.9T + 961T^{2} $
	37	$ 1 - 38.6T + 1.36e3T^{2} $
	41	$ 1 - 1.93iT - 1.68e3T^{2} $
	43	$ 1 - 52.6T + 1.84e3T^{2} $
	47	$ 1 - 2.59iT - 2.20e3T^{2} $
	53	$ 1 + 14.7iT - 2.80e3T^{2} $
	59	$ 1 + 17.0iT - 3.48e3T^{2} $
	61	$ 1 + 23.0T + 3.72e3T^{2} $
	67	$ 1 + 54.1T + 4.48e3T^{2} $
	71	$ 1 - 85.3iT - 5.04e3T^{2} $
	73	$ 1 - 3.47T + 5.32e3T^{2} $
	79	$ 1 - 16.2T + 6.24e3T^{2} $
	83	$ 1 + 92.0iT - 6.88e3T^{2} $
	89	$ 1 + 64.3iT - 7.92e3T^{2} $
	97	$ 1 - 44.3T + 9.40e3T^{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

−8.596770896361723574948366257253, −7.83035275870765341754747293289, −6.66864159317074479689404875368, −5.94243152511661109996901473625, −5.66825239712266321432215947072, −4.32606330598543392007894234019, −3.51498545436997877539383781802, −2.95304591963553510715024621975, −1.26694925286907457716426210272, −0.54490411895446741730187007290, 0.890485853471025990818010582693, 2.49685148422591968961403681137, 3.03489933038683812445382297852, 3.83541335890810512866615086272, 4.89466531086584872991720284774, 5.95154385836884461432065029670, 6.56917004587300987693156645524, 7.27263781201983272653036444458, 7.61523929798756033843138795156, 9.154338557927882134482895400594

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.717499038\)
\(L(\frac12)\)	\(\approx\)	\(1.717499038\)
\(L(2)\)		not available
\(L(1)\)		not available

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