Properties

Label 2-2e8-8.5-c5-0-33
Degree $2$
Conductor $256$
Sign $-0.707 + 0.707i$
Analytic cond. $41.0582$
Root an. cond. $6.40767$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12i·3-s − 54i·5-s + 88·7-s + 99·9-s − 540i·11-s − 418i·13-s − 648·15-s + 594·17-s + 836i·19-s − 1.05e3i·21-s + 4.10e3·23-s + 209·25-s − 4.10e3i·27-s − 594i·29-s + 4.25e3·31-s + ⋯
L(s)  = 1  − 0.769i·3-s − 0.965i·5-s + 0.678·7-s + 0.407·9-s − 1.34i·11-s − 0.685i·13-s − 0.743·15-s + 0.498·17-s + 0.531i·19-s − 0.522i·21-s + 1.61·23-s + 0.0668·25-s − 1.08i·27-s − 0.131i·29-s + 0.795·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(41.0582\)
Root analytic conductor: \(6.40767\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :5/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.341613047\)
\(L(\frac12)\) \(\approx\) \(2.341613047\)
\(L(\frac{7}{2})\) 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 \( 1 + 12iT - 243T^{2} \)
5 \( 1 + 54iT - 3.12e3T^{2} \)
7 \( 1 - 88T + 1.68e4T^{2} \)
11 \( 1 + 540iT - 1.61e5T^{2} \)
13 \( 1 + 418iT - 3.71e5T^{2} \)
17 \( 1 - 594T + 1.41e6T^{2} \)
19 \( 1 - 836iT - 2.47e6T^{2} \)
23 \( 1 - 4.10e3T + 6.43e6T^{2} \)
29 \( 1 + 594iT - 2.05e7T^{2} \)
31 \( 1 - 4.25e3T + 2.86e7T^{2} \)
37 \( 1 - 298iT - 6.93e7T^{2} \)
41 \( 1 + 1.72e4T + 1.15e8T^{2} \)
43 \( 1 - 1.21e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.29e3T + 2.29e8T^{2} \)
53 \( 1 + 1.94e4iT - 4.18e8T^{2} \)
59 \( 1 - 7.66e3iT - 7.14e8T^{2} \)
61 \( 1 + 3.47e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.18e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.68e4T + 1.80e9T^{2} \)
73 \( 1 + 6.75e4T + 2.07e9T^{2} \)
79 \( 1 + 7.69e4T + 3.07e9T^{2} \)
83 \( 1 - 6.77e4iT - 3.93e9T^{2} \)
89 \( 1 + 2.97e4T + 5.58e9T^{2} \)
97 \( 1 + 1.22e5T + 8.58e9T^{2} \)
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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

−10.96673501532699991965900545132, −9.800052300470994582882185240025, −8.463440606133406618673717390408, −8.116790953614393569051454657527, −6.85982653732257161847051811756, −5.63731839674385602910431264745, −4.71642954700158013437737952717, −3.16123620635294066555346658567, −1.42841803377364697087145988828, −0.73608196413052327172784557268, 1.58855452875001172237472826631, 2.99849158018346843442178430617, 4.34721650962262878894440722553, 5.07680719835353065028031038376, 6.81825283350181482479339067061, 7.31579272115181977647119822190, 8.801830771852822067513318981193, 9.800847885907381303845183186686, 10.49512921935053364460954883036, 11.30640195795238470713041236165

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