LMFDB - L-function 2-54e2-3.2-c2-0-71

Dirichlet series

L(s) = 1

− 7.21i·5^-s − 4.39·7^-s − 6.95i·11^-s − 20.9·13^-s − 25.1i·17^-s − 23.2·19^-s − 26.9i·23^-s − 27.0·25^-s + 8.80i·29^-s − 31.0·31^-s + 31.6i·35^-s + 27.8·37^-s + 30.0i·41^-s + 47.1·43^-s + 51.4i·47^-s + ⋯

L(s) = 1

− 1.44i·5^-s − 0.627·7^-s − 0.632i·11^-s − 1.61·13^-s − 1.47i·17^-s − 1.22·19^-s − 1.17i·23^-s − 1.08·25^-s + 0.303i·29^-s − 1.00·31^-s + 0.905i·35^-s + 0.753·37^-s + 0.732i·41^-s + 1.09·43^-s + 1.09i·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

$L(\frac{3}{2})$	$\approx$	$0.2060361703$
$L(\frac12)$	$\approx$	$0.2060361703$
$L(2)$		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 + 7.21iT - 25T^{2} $
	7	$ 1 + 4.39T + 49T^{2} $
	11	$ 1 + 6.95iT - 121T^{2} $
	13	$ 1 + 20.9T + 169T^{2} $
	17	$ 1 + 25.1iT - 289T^{2} $
	19	$ 1 + 23.2T + 361T^{2} $
	23	$ 1 + 26.9iT - 529T^{2} $
	29	$ 1 - 8.80iT - 841T^{2} $
	31	$ 1 + 31.0T + 961T^{2} $
	37	$ 1 - 27.8T + 1.36e3T^{2} $
	41	$ 1 - 30.0iT - 1.68e3T^{2} $
	43	$ 1 - 47.1T + 1.84e3T^{2} $
	47	$ 1 - 51.4iT - 2.20e3T^{2} $
	53	$ 1 + 37.5iT - 2.80e3T^{2} $
	59	$ 1 - 20.0iT - 3.48e3T^{2} $
	61	$ 1 - 97.3T + 3.72e3T^{2} $
	67	$ 1 + 74.6T + 4.48e3T^{2} $
	71	$ 1 + 59.2iT - 5.04e3T^{2} $
	73	$ 1 - 74.7T + 5.32e3T^{2} $
	79	$ 1 + 96.2T + 6.24e3T^{2} $
	83	$ 1 + 123. iT - 6.88e3T^{2} $
	89	$ 1 - 19.9iT - 7.92e3T^{2} $
	97	$ 1 + 12.2T + 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

−8.106193095692025617998637447069, −7.32576137913600094106365132181, −6.48376422418421952800729314783, −5.59340841573040739534864934422, −4.78791537052377844167128341017, −4.35779748738748246164246392149, −3.02964780122309553610534921835, −2.19393395000862043381490817779, −0.74895639214767211749160364596, −0.06077050015853974479511070258, 1.96212420651384662482390733419, 2.56183320236969231292700595757, 3.58833646304895632415267466326, 4.26285092734657883771249570437, 5.48826218935403018822669121275, 6.22711003576900637052165605126, 7.01252961873316961393546951293, 7.37259348818126687918113060232, 8.275610763482796689588843684985, 9.371111454334074581907334063282

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.2060361703\)
\(L(\frac12)\)	\(\approx\)	\(0.2060361703\)
\(L(2)\)		not available
\(L(1)\)		not available

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