LMFDB - L-function 2-72e2-24.11-c1-0-61

Dirichlet series

L(s) = 1

− 4.04·5^-s − 3.89i·7^-s − 0.468i·11^-s − 4.89i·13^-s + 3.57i·17^-s + 1.89·19^-s + 8.56·23^-s + 11.3·25^-s + 2.36·29^-s + 7.18i·31^-s + 15.7i·35^-s − 6.16i·37^-s + 5.13i·41^-s + 10.5·43^-s + 9.37·47^-s + ⋯

L(s) = 1

− 1.80·5^-s − 1.47i·7^-s − 0.141i·11^-s − 1.35i·13^-s + 0.867i·17^-s + 0.435·19^-s + 1.78·23^-s + 2.27·25^-s + 0.439·29^-s + 1.28i·31^-s + 2.66i·35^-s − 1.01i·37^-s + 0.801i·41^-s + 1.60·43^-s + 1.36·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5184$ = $2^{6} \cdot 3^{4}$
Sign:	$-0.258 + 0.965i$
Analytic conductor:	$41.3944$
Root analytic conductor:	$6.43385$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5184} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5184,\ (\ :1/2),\ -0.258 + 0.965i)$

Particular Values

$L(1)$	$\approx$	$1.207365279$
$L(\frac12)$	$\approx$	$1.207365279$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 4.04T + 5T^{2} $
	7	$ 1 + 3.89iT - 7T^{2} $
	11	$ 1 + 0.468iT - 11T^{2} $
	13	$ 1 + 4.89iT - 13T^{2} $
	17	$ 1 - 3.57iT - 17T^{2} $
	19	$ 1 - 1.89T + 19T^{2} $
	23	$ 1 - 8.56T + 23T^{2} $
	29	$ 1 - 2.36T + 29T^{2} $
	31	$ 1 - 7.18iT - 31T^{2} $
	37	$ 1 + 6.16iT - 37T^{2} $
	41	$ 1 - 5.13iT - 41T^{2} $
	43	$ 1 - 10.5T + 43T^{2} $
	47	$ 1 - 9.37T + 47T^{2} $
	53	$ 1 + 5.52T + 53T^{2} $
	59	$ 1 + 6.26iT - 59T^{2} $
	61	$ 1 + 0.983iT - 61T^{2} $
	67	$ 1 + 12.7T + 67T^{2} $
	71	$ 1 + 3.02T + 71T^{2} $
	73	$ 1 - 4.58T + 73T^{2} $
	79	$ 1 + 11.6iT - 79T^{2} $
	83	$ 1 + 5.66iT - 83T^{2} $
	89	$ 1 + 3.17iT - 89T^{2} $
	97	$ 1 - 0.611T + 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

−7.85721257715011684660510711288, −7.36129168322654849494480217895, −6.94971950865722089043472143043, −5.82553382416525840096459212707, −4.77475623542583602584378866617, −4.29363970084318007738201170241, −3.37661359673042173962479145489, −3.10357246695359276285301421355, −1.11492179233499569651245690538, −0.49011427702829569200575336644, 0.930778282707696304648133664816, 2.43356052410859814620402272882, 3.03572295049486504098484159583, 4.05820838972818613650462663454, 4.65740353604996455581799615885, 5.38339718781556568219493008679, 6.37576036582139058362824798678, 7.19750979248331002815809123508, 7.58041988330382049518912374097, 8.512170651979873138733725102999

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.207365279\)
\(L(\frac12)\)	\(\approx\)	\(1.207365279\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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