LMFDB - L-function 2-2e8-8.5-c5-0-26

Dirichlet series

L(s) = 1

− 20i·3^-s + 74i·5^-s − 24·7^-s − 157·9^-s + 124i·11^-s + 478i·13^-s + 1.48e3·15^-s − 1.19e3·17^-s − 3.04e3i·19^-s + 480i·21^-s + 184·23^-s − 2.35e3·25^-s − 1.72e3i·27^-s − 3.28e3i·29^-s + 5.72e3·31^-s + ⋯

L(s) = 1

− 1.28i·3^-s + 1.32i·5^-s − 0.185·7^-s − 0.646·9^-s + 0.308i·11^-s + 0.784i·13^-s + 1.69·15^-s − 1.00·17^-s − 1.93i·19^-s + 0.237i·21^-s + 0.0725·23^-s − 0.752·25^-s − 0.454i·27^-s − 0.724i·29^-s + 1.07·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$256$ = $2^{8}$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$41.0582$
Root analytic conductor:	$6.40767$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{256} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 256,\ (\ :5/2),\ -0.707 + 0.707i)$

Particular Values

$L(3)$	$\approx$	$1.092727895$
$L(\frac12)$	$\approx$	$1.092727895$
$L(\frac{7}{2})$		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 + 20iT - 243T^{2} $
	5	$ 1 - 74iT - 3.12e3T^{2} $
	7	$ 1 + 24T + 1.68e4T^{2} $
	11	$ 1 - 124iT - 1.61e5T^{2} $
	13	$ 1 - 478iT - 3.71e5T^{2} $
	17	$ 1 + 1.19e3T + 1.41e6T^{2} $
	19	$ 1 + 3.04e3iT - 2.47e6T^{2} $
	23	$ 1 - 184T + 6.43e6T^{2} $
	29	$ 1 + 3.28e3iT - 2.05e7T^{2} $
	31	$ 1 - 5.72e3T + 2.86e7T^{2} $
	37	$ 1 + 1.03e4iT - 6.93e7T^{2} $
	41	$ 1 - 8.88e3T + 1.15e8T^{2} $
	43	$ 1 + 9.18e3iT - 1.47e8T^{2} $
	47	$ 1 + 2.36e4T + 2.29e8T^{2} $
	53	$ 1 + 1.16e4iT - 4.18e8T^{2} $
	59	$ 1 - 1.68e4iT - 7.14e8T^{2} $
	61	$ 1 + 1.84e4iT - 8.44e8T^{2} $
	67	$ 1 - 1.55e4iT - 1.35e9T^{2} $
	71	$ 1 + 3.19e4T + 1.80e9T^{2} $
	73	$ 1 - 4.88e3T + 2.07e9T^{2} $
	79	$ 1 + 4.45e4T + 3.07e9T^{2} $
	83	$ 1 + 6.73e4iT - 3.93e9T^{2} $
	89	$ 1 + 7.19e4T + 5.58e9T^{2} $
	97	$ 1 - 4.88e4T + 8.58e9T^{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

−11.11978352593331766964150968316, −9.886029529969304667063226540731, −8.748820503442784800962148963092, −7.42771071193203155865829207830, −6.84726858017668134113999085001, −6.28000939179846108232812767170, −4.47251497737943694028708211884, −2.81664970818251678839883606371, −2.00778477468551833986573813147, −0.31672458671678899582905607413, 1.27978740445730790762136059629, 3.26016466579534332080095697165, 4.37991032047716966787286406760, 5.10600479756497898249165081885, 6.19369643979329962808526480953, 8.016298864661784869699502929620, 8.719924864879738623924814177993, 9.685606024860844524266183740564, 10.32474364038249819197637538675, 11.40307937028845557837432240109

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.092727895\)
\(L(\frac12)\)	\(\approx\)	\(1.092727895\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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