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Properties

Label 2-2e8-8.5-c7-0-9
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  + 14.3i·3-s + 167. i·5-s − 1.35e3·7-s + 1.97e3·9-s + 1.09e3i·11-s − 8.91e3i·13-s − 2.41e3·15-s + 2.02e4·17-s + 2.72e4i·19-s − 1.95e4i·21-s + 5.58e4·23-s + 5.00e4·25-s + 5.99e4i·27-s − 3.85e4i·29-s + 1.31e4·31-s + ⋯
L(s)  = 1  + 0.307i·3-s + 0.599i·5-s − 1.49·7-s + 0.905·9-s + 0.247i·11-s − 1.12i·13-s − 0.184·15-s + 1.00·17-s + 0.910i·19-s − 0.459i·21-s + 0.956·23-s + 0.640·25-s + 0.586i·27-s − 0.293i·29-s + 0.0793·31-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

\(L(4)\) \(\approx\) \(1.270762901\)
\(L(\frac12)\) \(\approx\) \(1.270762901\)
\(L(\frac{9}{2})\) 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 \( 1 - 14.3iT - 2.18e3T^{2} \)
5 \( 1 - 167. iT - 7.81e4T^{2} \)
7 \( 1 + 1.35e3T + 8.23e5T^{2} \)
11 \( 1 - 1.09e3iT - 1.94e7T^{2} \)
13 \( 1 + 8.91e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.02e4T + 4.10e8T^{2} \)
19 \( 1 - 2.72e4iT - 8.93e8T^{2} \)
23 \( 1 - 5.58e4T + 3.40e9T^{2} \)
29 \( 1 + 3.85e4iT - 1.72e10T^{2} \)
31 \( 1 - 1.31e4T + 2.75e10T^{2} \)
37 \( 1 - 4.36e5iT - 9.49e10T^{2} \)
41 \( 1 + 5.60e5T + 1.94e11T^{2} \)
43 \( 1 - 2.57e5iT - 2.71e11T^{2} \)
47 \( 1 + 7.39e5T + 5.06e11T^{2} \)
53 \( 1 - 5.91e5iT - 1.17e12T^{2} \)
59 \( 1 - 2.19e6iT - 2.48e12T^{2} \)
61 \( 1 + 3.09e6iT - 3.14e12T^{2} \)
67 \( 1 + 4.78e6iT - 6.06e12T^{2} \)
71 \( 1 - 1.98e6T + 9.09e12T^{2} \)
73 \( 1 - 4.99e5T + 1.10e13T^{2} \)
79 \( 1 + 3.74e6T + 1.92e13T^{2} \)
83 \( 1 - 9.51e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.12e7T + 4.42e13T^{2} \)
97 \( 1 + 8.93e6T + 8.07e13T^{2} \)
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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

−10.83016031556112106958103910048, −10.00561219945520339616948480409, −9.666754483505636442800154475682, −8.141808326390209112270182330746, −7.06099074114075923415995218697, −6.28344501561235667349163645827, −5.04808978761419657874590484852, −3.54615123134967641510867351732, −2.97329874836875791402702528814, −1.15459606380005898119085108960, 0.32987847842482799127455324272, 1.43580864572940945758406538997, 2.95955437296888206540445107384, 4.10633845044482324021904062705, 5.32905331406845319622805774658, 6.67608397659675631370726735235, 7.13719688419385667823961699570, 8.671496820736213053822468396595, 9.434437711849500746060099677231, 10.21006768314972932076751670280

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