LMFDB - L-function 2-54e2-3.2-c2-0-61

Dirichlet series

L(s) = 1

− 4.17i·5^-s + 5.19·7^-s − 7.03i·11^-s + 19.1·13^-s − 23.5i·17^-s + 23.5·19^-s − 23.8i·23^-s + 7.53·25^-s − 21.5i·29^-s − 25.5·31^-s − 21.7i·35^-s + 13.6·37^-s + 51.5i·41^-s + 4.64·43^-s + 4.80i·47^-s + ⋯

L(s) = 1

− 0.835i·5^-s + 0.742·7^-s − 0.639i·11^-s + 1.47·13^-s − 1.38i·17^-s + 1.24·19^-s − 1.03i·23^-s + 0.301·25^-s − 0.742i·29^-s − 0.824·31^-s − 0.620i·35^-s + 0.368·37^-s + 1.25i·41^-s + 0.107·43^-s + 0.102i·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

$L(\frac{3}{2})$	$\approx$	$2.661425892$
$L(\frac12)$	$\approx$	$2.661425892$
$L(2)$		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 + 4.17iT - 25T^{2} $
	7	$ 1 - 5.19T + 49T^{2} $
	11	$ 1 + 7.03iT - 121T^{2} $
	13	$ 1 - 19.1T + 169T^{2} $
	17	$ 1 + 23.5iT - 289T^{2} $
	19	$ 1 - 23.5T + 361T^{2} $
	23	$ 1 + 23.8iT - 529T^{2} $
	29	$ 1 + 21.5iT - 841T^{2} $
	31	$ 1 + 25.5T + 961T^{2} $
	37	$ 1 - 13.6T + 1.36e3T^{2} $
	41	$ 1 - 51.5iT - 1.68e3T^{2} $
	43	$ 1 - 4.64T + 1.84e3T^{2} $
	47	$ 1 - 4.80iT - 2.20e3T^{2} $
	53	$ 1 - 67.3iT - 2.80e3T^{2} $
	59	$ 1 + 86.7iT - 3.48e3T^{2} $
	61	$ 1 + 53.5T + 3.72e3T^{2} $
	67	$ 1 - 20.1T + 4.48e3T^{2} $
	71	$ 1 - 98.4iT - 5.04e3T^{2} $
	73	$ 1 - 139.T + 5.32e3T^{2} $
	79	$ 1 - 105.T + 6.24e3T^{2} $
	83	$ 1 - 80.4iT - 6.88e3T^{2} $
	89	$ 1 + 67.0iT - 7.92e3T^{2} $
	97	$ 1 + 146.T + 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

−8.257730767061955017993532943790, −7.965569050197452503947154646105, −6.86963962240408305313080622786, −6.01211390113011738723560751988, −5.20919594016726561806515688404, −4.63415794136947094430326721974, −3.63811045237410062409370599693, −2.68177472923519813944585811847, −1.32064348681867802414354570498, −0.69372086760216446633206496417, 1.24860686284103280647756083099, 1.99125075936861856578566507188, 3.35610697613483203018685733705, 3.79712113356742157800332981202, 4.98798856002466876609957856099, 5.73972285210452321554623390393, 6.52904308597310315793852191506, 7.32548844126799884980715072775, 7.931500815724274732114311353378, 8.740402322236485303040936601870

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.661425892\)
\(L(\frac12)\)	\(\approx\)	\(2.661425892\)
\(L(2)\)		not available
\(L(1)\)		not available

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