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Properties

Label 2-2e8-16.5-c3-0-5
Degree $2$
Conductor $256$
Sign $-0.793 + 0.608i$
Analytic cond. $15.1044$
Root an. cond. $3.88644$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.63 + 5.63i)3-s + (−0.329 − 0.329i)5-s + 17.2i·7-s − 36.4i·9-s + (44.3 + 44.3i)11-s + (−57.9 + 57.9i)13-s + 3.71·15-s + 13.0·17-s + (−68.4 + 68.4i)19-s + (−96.8 − 96.8i)21-s − 119. i·23-s − 124. i·25-s + (53.2 + 53.2i)27-s + (135. − 135. i)29-s − 40.6·31-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)3-s + (−0.0294 − 0.0294i)5-s + 0.928i·7-s − 1.34i·9-s + (1.21 + 1.21i)11-s + (−1.23 + 1.23i)13-s + 0.0639·15-s + 0.186·17-s + (−0.826 + 0.826i)19-s + (−1.00 − 1.00i)21-s − 1.08i·23-s − 0.998i·25-s + (0.379 + 0.379i)27-s + (0.866 − 0.866i)29-s − 0.235·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.793 + 0.608i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ -0.793 + 0.608i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5649968229\)
\(L(\frac12)\) \(\approx\) \(0.5649968229\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (5.63 - 5.63i)T - 27iT^{2} \)
5 \( 1 + (0.329 + 0.329i)T + 125iT^{2} \)
7 \( 1 - 17.2iT - 343T^{2} \)
11 \( 1 + (-44.3 - 44.3i)T + 1.33e3iT^{2} \)
13 \( 1 + (57.9 - 57.9i)T - 2.19e3iT^{2} \)
17 \( 1 - 13.0T + 4.91e3T^{2} \)
19 \( 1 + (68.4 - 68.4i)T - 6.85e3iT^{2} \)
23 \( 1 + 119. iT - 1.21e4T^{2} \)
29 \( 1 + (-135. + 135. i)T - 2.43e4iT^{2} \)
31 \( 1 + 40.6T + 2.97e4T^{2} \)
37 \( 1 + (130. + 130. i)T + 5.06e4iT^{2} \)
41 \( 1 - 283. iT - 6.89e4T^{2} \)
43 \( 1 + (-48.9 - 48.9i)T + 7.95e4iT^{2} \)
47 \( 1 + 226.T + 1.03e5T^{2} \)
53 \( 1 + (471. + 471. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-81.1 - 81.1i)T + 2.05e5iT^{2} \)
61 \( 1 + (14.7 - 14.7i)T - 2.26e5iT^{2} \)
67 \( 1 + (199. - 199. i)T - 3.00e5iT^{2} \)
71 \( 1 - 678. iT - 3.57e5T^{2} \)
73 \( 1 - 26.2iT - 3.89e5T^{2} \)
79 \( 1 + 720.T + 4.93e5T^{2} \)
83 \( 1 + (477. - 477. i)T - 5.71e5iT^{2} \)
89 \( 1 - 508. iT - 7.04e5T^{2} \)
97 \( 1 - 1.13e3T + 9.12e5T^{2} \)
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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

−12.13807352456699680601749812348, −11.32968141896676147276310016687, −10.00843230610284950814020954302, −9.676008115124781492365831686882, −8.538619505674836194269434442331, −6.83103891561427193497738553919, −6.06160646953480972443307525921, −4.69831077757326371321917950352, −4.26064130470624898988933010077, −2.13415318112629730607024616211, 0.27408569135929623095737637633, 1.31131738951338892625920083896, 3.33991212787095657403048155409, 4.99708102608959688490108691805, 6.03248842219868111291499439396, 6.98585832967913446244475407843, 7.64273898769381979247448524316, 8.997728851086121763120434746112, 10.39918820228040179953629083897, 11.13319363353160874693397835460

Graph of the $Z$-function along the critical line