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Properties

Label 2-2e8-4.3-c10-0-30
Degree $2$
Conductor $256$
Sign $-i$
Analytic cond. $162.651$
Root an. cond. $12.7534$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 196. i·3-s − 5.38e3·5-s + 6.61e3i·7-s + 2.03e4·9-s + 2.21e5i·11-s − 3.61e5·13-s − 1.05e6i·15-s + 7.76e5·17-s − 1.46e6i·19-s − 1.30e6·21-s − 2.34e6i·23-s + 1.91e7·25-s + 1.56e7i·27-s + 2.88e7·29-s − 3.00e7i·31-s + ⋯
L(s)  = 1  + 0.809i·3-s − 1.72·5-s + 0.393i·7-s + 0.344·9-s + 1.37i·11-s − 0.974·13-s − 1.39i·15-s + 0.546·17-s − 0.590i·19-s − 0.318·21-s − 0.364i·23-s + 1.96·25-s + 1.08i·27-s + 1.40·29-s − 1.05i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-i$
Analytic conductor: \(162.651\)
Root analytic conductor: \(12.7534\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :5),\ -i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.361272030\)
\(L(\frac12)\) \(\approx\) \(1.361272030\)
\(L(6)\) 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 \( 1 - 196. iT - 5.90e4T^{2} \)
5 \( 1 + 5.38e3T + 9.76e6T^{2} \)
7 \( 1 - 6.61e3iT - 2.82e8T^{2} \)
11 \( 1 - 2.21e5iT - 2.59e10T^{2} \)
13 \( 1 + 3.61e5T + 1.37e11T^{2} \)
17 \( 1 - 7.76e5T + 2.01e12T^{2} \)
19 \( 1 + 1.46e6iT - 6.13e12T^{2} \)
23 \( 1 + 2.34e6iT - 4.14e13T^{2} \)
29 \( 1 - 2.88e7T + 4.20e14T^{2} \)
31 \( 1 + 3.00e7iT - 8.19e14T^{2} \)
37 \( 1 - 1.16e7T + 4.80e15T^{2} \)
41 \( 1 + 4.70e7T + 1.34e16T^{2} \)
43 \( 1 - 3.52e7iT - 2.16e16T^{2} \)
47 \( 1 + 4.23e8iT - 5.25e16T^{2} \)
53 \( 1 + 9.16e6T + 1.74e17T^{2} \)
59 \( 1 + 1.03e9iT - 5.11e17T^{2} \)
61 \( 1 - 1.37e8T + 7.13e17T^{2} \)
67 \( 1 - 7.94e8iT - 1.82e18T^{2} \)
71 \( 1 - 2.04e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.88e9T + 4.29e18T^{2} \)
79 \( 1 - 1.04e9iT - 9.46e18T^{2} \)
83 \( 1 + 3.77e9iT - 1.55e19T^{2} \)
89 \( 1 - 4.07e9T + 3.11e19T^{2} \)
97 \( 1 - 1.76e9T + 7.37e19T^{2} \)
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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

−10.34763777829087937925970535042, −9.686055445866027568374382589412, −8.543883143677249093485152162997, −7.53970350758911409945024409520, −6.89735957176571764179179220024, −4.98692936245681909748085666852, −4.48259086723902539996395721335, −3.57902089694103589323532079537, −2.33617822876110823003163363797, −0.62063746327963536036151726910, 0.49402220785405352201987440688, 1.17701926769210835429536138166, 2.89515428881335257044735160417, 3.79343905523899950834281468177, 4.82970061298905928775106806379, 6.31492535084726759494592887246, 7.36462284412013147450740360793, 7.82847596887617003752384582225, 8.684674619584195574282538738681, 10.19304687240211115837722658210

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