LMFDB - L-function 2-2e8-8.5-c3-0-3

Dirichlet series

L(s) = 1

− 8.92i·3^-s + 11.8i·5^-s + 9.85·7^-s − 52.7·9^-s + 39.0i·11^-s + 91.5i·13^-s + 105.·15^-s − 37.1·17^-s + 46.4i·19^-s − 87.9i·21^-s − 120.·23^-s − 15.5·25^-s + 229. i·27^-s − 27.2i·29^-s + 81.1·31^-s + ⋯

L(s) = 1

− 1.71i·3^-s + 1.06i·5^-s + 0.532·7^-s − 1.95·9^-s + 1.07i·11^-s + 1.95i·13^-s + 1.82·15^-s − 0.529·17^-s + 0.561i·19^-s − 0.914i·21^-s − 1.09·23^-s − 0.124·25^-s + 1.63i·27^-s − 0.174i·29^-s + 0.470·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$256$ = $2^{8}$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$15.1044$
Root analytic conductor:	$3.88644$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{256} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 256,\ (\ :3/2),\ 0.707 - 0.707i)$

Particular Values

$L(2)$	$\approx$	$1.356357592$
$L(\frac12)$	$\approx$	$1.356357592$
$L(\frac{5}{2})$		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 + 8.92iT - 27T^{2} $
	5	$ 1 - 11.8iT - 125T^{2} $
	7	$ 1 - 9.85T + 343T^{2} $
	11	$ 1 - 39.0iT - 1.33e3T^{2} $
	13	$ 1 - 91.5iT - 2.19e3T^{2} $
	17	$ 1 + 37.1T + 4.91e3T^{2} $
	19	$ 1 - 46.4iT - 6.85e3T^{2} $
	23	$ 1 + 120.T + 1.21e4T^{2} $
	29	$ 1 + 27.2iT - 2.43e4T^{2} $
	31	$ 1 - 81.1T + 2.97e4T^{2} $
	37	$ 1 + 10.9iT - 5.06e4T^{2} $
	41	$ 1 - 205.T + 6.89e4T^{2} $
	43	$ 1 - 115. iT - 7.95e4T^{2} $
	47	$ 1 - 312.T + 1.03e5T^{2} $
	53	$ 1 + 90.9iT - 1.48e5T^{2} $
	59	$ 1 + 550. iT - 2.05e5T^{2} $
	61	$ 1 - 630. iT - 2.26e5T^{2} $
	67	$ 1 - 661. iT - 3.00e5T^{2} $
	71	$ 1 - 494.T + 3.57e5T^{2} $
	73	$ 1 + 566.T + 3.89e5T^{2} $
	79	$ 1 + 49.4T + 4.93e5T^{2} $
	83	$ 1 + 564. iT - 5.71e5T^{2} $
	89	$ 1 + 1.08e3T + 7.04e5T^{2} $
	97	$ 1 - 464.T + 9.12e5T^{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

−11.79069412700240623532499080992, −11.11413444751993304921168885904, −9.775172688770889438716953149317, −8.489600163539730666309615357752, −7.43251886800431269972583550192, −6.87113830797118259727467633018, −6.10409410486400240664974232508, −4.30436513088479533520626670424, −2.40611688742016800042345132177, −1.67046399666799519507505707816, 0.53076736711054078274571203394, 3.01124314633498597498321201195, 4.21474652664409196925636977490, 5.14066812895528069451341746572, 5.81959117671238354856040867616, 8.064819607428414646995240713303, 8.626523950069466747739106971347, 9.537954010153947817844661160309, 10.56213918613867495800334457314, 11.09941572731433435110178781052

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.356357592\)
\(L(\frac12)\)	\(\approx\)	\(1.356357592\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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