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Properties

Label 2-2e8-8.5-c7-0-40
Degree $2$
Conductor $256$
Sign $0.707 - 0.707i$
Analytic cond. $79.9705$
Root an. cond. $8.94262$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 62.2i·3-s + 203. i·5-s + 534.·7-s − 1.68e3·9-s − 1.57e3i·11-s − 1.77e3i·13-s − 1.26e4·15-s + 2.89e4·17-s − 3.92e4i·19-s + 3.32e4i·21-s + 8.52e4·23-s + 3.69e4·25-s + 3.09e4i·27-s − 2.46e5i·29-s + 5.46e4·31-s + ⋯
L(s)  = 1  + 1.33i·3-s + 0.726i·5-s + 0.589·7-s − 0.772·9-s − 0.357i·11-s − 0.223i·13-s − 0.966·15-s + 1.43·17-s − 1.31i·19-s + 0.784i·21-s + 1.46·23-s + 0.472·25-s + 0.303i·27-s − 1.87i·29-s + 0.329·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(79.9705\)
Root analytic conductor: \(8.94262\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :7/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.619577539\)
\(L(\frac12)\) \(\approx\) \(2.619577539\)
\(L(\frac{9}{2})\) 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 \( 1 - 62.2iT - 2.18e3T^{2} \)
5 \( 1 - 203. iT - 7.81e4T^{2} \)
7 \( 1 - 534.T + 8.23e5T^{2} \)
11 \( 1 + 1.57e3iT - 1.94e7T^{2} \)
13 \( 1 + 1.77e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.89e4T + 4.10e8T^{2} \)
19 \( 1 + 3.92e4iT - 8.93e8T^{2} \)
23 \( 1 - 8.52e4T + 3.40e9T^{2} \)
29 \( 1 + 2.46e5iT - 1.72e10T^{2} \)
31 \( 1 - 5.46e4T + 2.75e10T^{2} \)
37 \( 1 + 2.90e5iT - 9.49e10T^{2} \)
41 \( 1 - 7.25e5T + 1.94e11T^{2} \)
43 \( 1 + 6.68e5iT - 2.71e11T^{2} \)
47 \( 1 + 4.29e5T + 5.06e11T^{2} \)
53 \( 1 + 7.25e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.65e5iT - 2.48e12T^{2} \)
61 \( 1 + 2.66e6iT - 3.14e12T^{2} \)
67 \( 1 - 1.16e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.87e6T + 9.09e12T^{2} \)
73 \( 1 + 5.79e5T + 1.10e13T^{2} \)
79 \( 1 + 1.92e6T + 1.92e13T^{2} \)
83 \( 1 - 4.86e6iT - 2.71e13T^{2} \)
89 \( 1 + 4.58e6T + 4.42e13T^{2} \)
97 \( 1 + 4.18e6T + 8.07e13T^{2} \)
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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

−10.97125775257670835952716741243, −9.969238398440249757532582160607, −9.218120877931517428057161772983, −8.068709081733376495561147328475, −6.95656761411021750135330517477, −5.57622454643855621196580708641, −4.71838532335857996626725777808, −3.58385466286082233194605839874, −2.63418767984910444257120306746, −0.72956890728057324069030438090, 1.18734159288340735182814383141, 1.36891822380110051748791910257, 3.00159384701357200939345191040, 4.64618018801703275784393023451, 5.65203757594413876791201718042, 6.85116287923830047890835858442, 7.73005869623574463165042967939, 8.425162619692323596123237835329, 9.559523454494046794510419383539, 10.80367760955281309347884828781

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