LMFDB - L-function 2-2e8-8.5-c7-0-5

Dirichlet series

L(s) = 1

− 58i·5^-s + 2.18e3·9^-s + 8.89e3i·13^-s − 4.00e4·17^-s + 7.47e4·25^-s − 2.33e5i·29^-s + 5.63e5i·37^-s − 9.53e3·41^-s − 1.26e5i·45^-s − 8.23e5·49^-s − 7.98e5i·53^-s + 3.50e6i·61^-s + 5.16e5·65^-s − 3.91e6·73^-s + 4.78e6·81^-s + ⋯

L(s) = 1

− 0.207i·5^-s + 9^-s + 1.12i·13^-s − 1.97·17^-s + 0.956·25^-s − 1.77i·29^-s + 1.83i·37^-s − 0.0215·41^-s − 0.207i·45^-s − 49^-s − 0.736i·53^-s + 1.97i·61^-s + 0.233·65^-s − 1.17·73^-s + 81^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$256$ = $2^{8}$
Sign:	$-0.707 - 0.707i$
Analytic conductor:	$79.9705$
Root analytic conductor:	$8.94262$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{256} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 256,\ (\ :7/2),\ -0.707 - 0.707i)$

Particular Values

$L(4)$	$\approx$	$0.8588077309$
$L(\frac12)$	$\approx$	$0.8588077309$
$L(\frac{9}{2})$		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 $
good	3	$ 1 - 2.18e3T^{2} $
	5	$ 1 + 58iT - 7.81e4T^{2} $
	7	$ 1 + 8.23e5T^{2} $
	11	$ 1 - 1.94e7T^{2} $
	13	$ 1 - 8.89e3iT - 6.27e7T^{2} $
	17	$ 1 + 4.00e4T + 4.10e8T^{2} $
	19	$ 1 - 8.93e8T^{2} $
	23	$ 1 + 3.40e9T^{2} $
	29	$ 1 + 2.33e5iT - 1.72e10T^{2} $
	31	$ 1 + 2.75e10T^{2} $
	37	$ 1 - 5.63e5iT - 9.49e10T^{2} $
	41	$ 1 + 9.53e3T + 1.94e11T^{2} $
	43	$ 1 - 2.71e11T^{2} $
	47	$ 1 + 5.06e11T^{2} $
	53	$ 1 + 7.98e5iT - 1.17e12T^{2} $
	59	$ 1 - 2.48e12T^{2} $
	61	$ 1 - 3.50e6iT - 3.14e12T^{2} $
	67	$ 1 - 6.06e12T^{2} $
	71	$ 1 + 9.09e12T^{2} $
	73	$ 1 + 3.91e6T + 1.10e13T^{2} $
	79	$ 1 + 1.92e13T^{2} $
	83	$ 1 - 2.71e13T^{2} $
	89	$ 1 + 9.24e6T + 4.42e13T^{2} $
	97	$ 1 + 1.75e7T + 8.07e13T^{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

−11.20255807738571982125331653307, −10.10120348455262132812530448503, −9.229796821030971709167218716401, −8.323826884162768501970937251353, −7.01468541366574405450022340293, −6.38757839404935632178326071947, −4.73182339802749230894809410282, −4.14090996317143924029891382775, −2.43809156404570826930909614831, −1.32489436017951333899372613098, 0.19462660925242371691595585269, 1.59889341212149281161137643746, 2.90568590634306187181472596953, 4.19144476908664730311459225371, 5.22514309247016007453366243162, 6.59003793280927384686564897775, 7.31380105152670964941081467558, 8.527168978966713563428536029372, 9.441281789637447849265232830041, 10.62613301583465156320378916628

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(0.8588077309\)
\(L(\frac12)\)	\(\approx\)	\(0.8588077309\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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