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Properties

Label 2-2e8-8.5-c7-0-22
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  + 2.95i·3-s + 362. i·5-s + 715.·7-s + 2.17e3·9-s + 7.29e3i·11-s + 1.03e4i·13-s − 1.07e3·15-s − 1.12e4·17-s + 2.66e4i·19-s + 2.11e3i·21-s + 2.35e4·23-s − 5.32e4·25-s + 1.29e4i·27-s − 4.41e4i·29-s + 3.11e5·31-s + ⋯
L(s)  = 1  + 0.0632i·3-s + 1.29i·5-s + 0.787·7-s + 0.995·9-s + 1.65i·11-s + 1.31i·13-s − 0.0820·15-s − 0.554·17-s + 0.892i·19-s + 0.0498i·21-s + 0.403·23-s − 0.681·25-s + 0.126i·27-s − 0.336i·29-s + 1.87·31-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

\(L(4)\) \(\approx\) \(2.574599407\)
\(L(\frac12)\) \(\approx\) \(2.574599407\)
\(L(\frac{9}{2})\) 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 \( 1 - 2.95iT - 2.18e3T^{2} \)
5 \( 1 - 362. iT - 7.81e4T^{2} \)
7 \( 1 - 715.T + 8.23e5T^{2} \)
11 \( 1 - 7.29e3iT - 1.94e7T^{2} \)
13 \( 1 - 1.03e4iT - 6.27e7T^{2} \)
17 \( 1 + 1.12e4T + 4.10e8T^{2} \)
19 \( 1 - 2.66e4iT - 8.93e8T^{2} \)
23 \( 1 - 2.35e4T + 3.40e9T^{2} \)
29 \( 1 + 4.41e4iT - 1.72e10T^{2} \)
31 \( 1 - 3.11e5T + 2.75e10T^{2} \)
37 \( 1 + 1.39e5iT - 9.49e10T^{2} \)
41 \( 1 - 7.53e5T + 1.94e11T^{2} \)
43 \( 1 + 1.25e5iT - 2.71e11T^{2} \)
47 \( 1 + 7.98e5T + 5.06e11T^{2} \)
53 \( 1 + 1.14e6iT - 1.17e12T^{2} \)
59 \( 1 + 2.01e6iT - 2.48e12T^{2} \)
61 \( 1 + 5.67e3iT - 3.14e12T^{2} \)
67 \( 1 - 3.80e6iT - 6.06e12T^{2} \)
71 \( 1 + 2.78e6T + 9.09e12T^{2} \)
73 \( 1 - 1.85e6T + 1.10e13T^{2} \)
79 \( 1 - 2.26e6T + 1.92e13T^{2} \)
83 \( 1 - 2.79e6iT - 2.71e13T^{2} \)
89 \( 1 - 5.86e6T + 4.42e13T^{2} \)
97 \( 1 + 1.46e7T + 8.07e13T^{2} \)
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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

−11.09857758586232522878554336648, −10.13765306690695915540846706748, −9.517225799289554955174642319700, −7.987614983584073373409662930908, −7.06020942298229671350346304421, −6.51994879927847505526180830894, −4.73901450322078396936279969653, −4.04674004351708643702997407621, −2.38120768253847334088560743344, −1.57124592491336893850328830227, 0.66532081652600839736248434986, 1.19490742575214262605499157746, 2.91048497680093999802850724531, 4.42879412310107096566565248408, 5.12892760076925434994803229187, 6.27003524556619454301495101365, 7.77487524322294404414558959711, 8.428542925613757742096608694975, 9.249848958198717120275813650453, 10.53081743578753968471103692828

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