LMFDB - L-function 2-2e8-8.5-c7-0-44

Dirichlet series

L(s) = 1

− 61.9i·3^-s − 70i·5^-s + 1.11e3·7^-s − 1.65e3·9^-s − 7.25e3i·11^-s + 1.37e4i·13^-s − 4.33e3·15^-s + 1.69e4·17^-s − 3.40e4i·19^-s − 6.91e4i·21^-s − 3.23e4·23^-s + 7.32e4·25^-s − 3.30e4i·27^-s + 3.41e4i·29^-s + 1.20e5·31^-s + ⋯

L(s) = 1

− 1.32i·3^-s − 0.250i·5^-s + 1.22·7^-s − 0.755·9^-s − 1.64i·11^-s + 1.73i·13^-s − 0.331·15^-s + 0.838·17^-s − 1.13i·19^-s − 1.62i·21^-s − 0.554·23^-s + 0.937·25^-s − 0.323i·27^-s + 0.260i·29^-s + 0.726·31^-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

$L(4)$	$\approx$	$2.637052717$
$L(\frac12)$	$\approx$	$2.637052717$
$L(\frac{9}{2})$		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 + 61.9iT - 2.18e3T^{2} $
	5	$ 1 + 70iT - 7.81e4T^{2} $
	7	$ 1 - 1.11e3T + 8.23e5T^{2} $
	11	$ 1 + 7.25e3iT - 1.94e7T^{2} $
	13	$ 1 - 1.37e4iT - 6.27e7T^{2} $
	17	$ 1 - 1.69e4T + 4.10e8T^{2} $
	19	$ 1 + 3.40e4iT - 8.93e8T^{2} $
	23	$ 1 + 3.23e4T + 3.40e9T^{2} $
	29	$ 1 - 3.41e4iT - 1.72e10T^{2} $
	31	$ 1 - 1.20e5T + 2.75e10T^{2} $
	37	$ 1 + 3.52e4iT - 9.49e10T^{2} $
	41	$ 1 - 4.84e5T + 1.94e11T^{2} $
	43	$ 1 + 6.72e5iT - 2.71e11T^{2} $
	47	$ 1 - 1.20e6T + 5.06e11T^{2} $
	53	$ 1 + 8.51e5iT - 1.17e12T^{2} $
	59	$ 1 - 6.95e5iT - 2.48e12T^{2} $
	61	$ 1 - 7.16e4iT - 3.14e12T^{2} $
	67	$ 1 - 3.07e5iT - 6.06e12T^{2} $
	71	$ 1 + 7.57e5T + 9.09e12T^{2} $
	73	$ 1 + 3.91e6T + 1.10e13T^{2} $
	79	$ 1 + 3.14e5T + 1.92e13T^{2} $
	83	$ 1 - 1.53e6iT - 2.71e13T^{2} $
	89	$ 1 - 2.51e6T + 4.42e13T^{2} $
	97	$ 1 + 5.00e4T + 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

−10.75644760642620754390255811269, −9.034836387889244766467290691766, −8.407573536036333590665105782195, −7.45175219942817744235844411899, −6.55768104079033396778465101200, −5.47891933340203898073300112908, −4.22577373900865559810605861913, −2.55443152008596408274382217893, −1.43166718823538659867755436808, −0.67921037806725820929373863086, 1.28397260085064935782530966050, 2.77869418864528220908269244776, 4.08743163065655329897584938884, 4.86923045540551229176974608785, 5.74600138135242181585003679789, 7.51628677240496852986523660528, 8.141533779262990551494432528893, 9.516331743676035472637693145317, 10.29608874342175302898680261582, 10.71136938784929583662809113981

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(2.637052717\)
\(L(\frac12)\)	\(\approx\)	\(2.637052717\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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