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Properties

Label 2-2e8-4.3-c10-0-1
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  − 203. i·3-s − 2.08e3·5-s − 2.53e4i·7-s + 1.74e4·9-s + 1.32e5i·11-s + 3.04e5·13-s + 4.25e5i·15-s + 1.87e6·17-s + 2.20e6i·19-s − 5.17e6·21-s + 4.83e6i·23-s − 5.40e6·25-s − 1.56e7i·27-s − 2.08e7·29-s − 4.27e7i·31-s + ⋯
L(s)  = 1  − 0.838i·3-s − 0.668·5-s − 1.50i·7-s + 0.296·9-s + 0.822i·11-s + 0.818·13-s + 0.560i·15-s + 1.32·17-s + 0.889i·19-s − 1.26·21-s + 0.751i·23-s − 0.553·25-s − 1.08i·27-s − 1.01·29-s − 1.49i·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

\(L(\frac{11}{2})\) \(\approx\) \(0.2935226691\)
\(L(\frac12)\) \(\approx\) \(0.2935226691\)
\(L(6)\) 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 \( 1 + 203. iT - 5.90e4T^{2} \)
5 \( 1 + 2.08e3T + 9.76e6T^{2} \)
7 \( 1 + 2.53e4iT - 2.82e8T^{2} \)
11 \( 1 - 1.32e5iT - 2.59e10T^{2} \)
13 \( 1 - 3.04e5T + 1.37e11T^{2} \)
17 \( 1 - 1.87e6T + 2.01e12T^{2} \)
19 \( 1 - 2.20e6iT - 6.13e12T^{2} \)
23 \( 1 - 4.83e6iT - 4.14e13T^{2} \)
29 \( 1 + 2.08e7T + 4.20e14T^{2} \)
31 \( 1 + 4.27e7iT - 8.19e14T^{2} \)
37 \( 1 + 1.24e8T + 4.80e15T^{2} \)
41 \( 1 + 1.24e8T + 1.34e16T^{2} \)
43 \( 1 - 3.15e7iT - 2.16e16T^{2} \)
47 \( 1 - 2.91e7iT - 5.25e16T^{2} \)
53 \( 1 - 5.07e8T + 1.74e17T^{2} \)
59 \( 1 - 1.64e8iT - 5.11e17T^{2} \)
61 \( 1 - 3.21e7T + 7.13e17T^{2} \)
67 \( 1 - 3.70e8iT - 1.82e18T^{2} \)
71 \( 1 + 3.20e8iT - 3.25e18T^{2} \)
73 \( 1 + 2.69e9T + 4.29e18T^{2} \)
79 \( 1 - 1.89e9iT - 9.46e18T^{2} \)
83 \( 1 - 2.20e9iT - 1.55e19T^{2} \)
89 \( 1 + 9.12e9T + 3.11e19T^{2} \)
97 \( 1 + 1.89e9T + 7.37e19T^{2} \)
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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

−10.40175546880758686990292102302, −9.759719940181203234269942526995, −8.100307158357961496537806762899, −7.51555663146837526192862352179, −6.95020645243128063828548630700, −5.64412016591122386145781307647, −4.09053664191816306161160407224, −3.62968901369074370885975369321, −1.73713515822345540484198721429, −1.07804200185026475527007328710, 0.05993398353628556758077672615, 1.54704132269650550019809365512, 3.05770220983280522592136174197, 3.74338307441142905269937450195, 5.06060912294177938397915245519, 5.76760859475464686315504554340, 7.08624858908380768356851537277, 8.482946649990152690615463223021, 8.858420012310826097212539516585, 10.02426561920534091753271766806

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