Properties

Degree $2$
Conductor $128$
Sign $-0.889 + 0.457i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.91 − 3.91i)3-s + (−4.72 − 4.72i)5-s + 45.3·7-s − 50.3i·9-s + (−110. + 110. i)11-s + (157. − 157. i)13-s + 36.9i·15-s − 378.·17-s + (−203. − 203. i)19-s + (−177. − 177. i)21-s − 740.·23-s − 580. i·25-s + (−514. + 514. i)27-s + (−82.6 + 82.6i)29-s + 286. i·31-s + ⋯
L(s)  = 1  + (−0.434 − 0.434i)3-s + (−0.188 − 0.188i)5-s + 0.925·7-s − 0.621i·9-s + (−0.910 + 0.910i)11-s + (0.929 − 0.929i)13-s + 0.164i·15-s − 1.31·17-s + (−0.562 − 0.562i)19-s + (−0.402 − 0.402i)21-s − 1.39·23-s − 0.928i·25-s + (−0.705 + 0.705i)27-s + (−0.0982 + 0.0982i)29-s + 0.297i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.889 + 0.457i$
Motivic weight: \(4\)
Character: $\chi_{128} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ -0.889 + 0.457i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.182937 - 0.755361i\)
\(L(\frac12)\) \(\approx\) \(0.182937 - 0.755361i\)
\(L(3)\) 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 \( 1 + (3.91 + 3.91i)T + 81iT^{2} \)
5 \( 1 + (4.72 + 4.72i)T + 625iT^{2} \)
7 \( 1 - 45.3T + 2.40e3T^{2} \)
11 \( 1 + (110. - 110. i)T - 1.46e4iT^{2} \)
13 \( 1 + (-157. + 157. i)T - 2.85e4iT^{2} \)
17 \( 1 + 378.T + 8.35e4T^{2} \)
19 \( 1 + (203. + 203. i)T + 1.30e5iT^{2} \)
23 \( 1 + 740.T + 2.79e5T^{2} \)
29 \( 1 + (82.6 - 82.6i)T - 7.07e5iT^{2} \)
31 \( 1 - 286. iT - 9.23e5T^{2} \)
37 \( 1 + (1.47e3 + 1.47e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 1.30e3iT - 2.82e6T^{2} \)
43 \( 1 + (-366. + 366. i)T - 3.41e6iT^{2} \)
47 \( 1 - 751. iT - 4.87e6T^{2} \)
53 \( 1 + (-1.92e3 - 1.92e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-1.35e3 + 1.35e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-1.83e3 + 1.83e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (2.20e3 + 2.20e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 8.97e3T + 2.54e7T^{2} \)
73 \( 1 - 9.35e3iT - 2.83e7T^{2} \)
79 \( 1 + 2.86e3iT - 3.89e7T^{2} \)
83 \( 1 + (-1.03e3 - 1.03e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 5.17e3iT - 6.27e7T^{2} \)
97 \( 1 - 8.53e3T + 8.85e7T^{2} \)
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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

−12.30554304586642228232433622589, −11.19725301906008736275098353580, −10.37035335503331641881317437444, −8.815366382738124865336619004861, −7.88271377060738214888572472096, −6.67616786795269967807145903143, −5.42220392615169268901238165214, −4.14123043348694948176047826125, −2.07143225294614463861629203057, −0.32788570262435370115277626504, 1.97710864263946820925695107590, 3.97229842411239499198710546147, 5.12177978617229329500345981411, 6.31181203454675091429182990711, 7.904951109907914974026352468307, 8.664607335048198235645718566363, 10.27415043199377014098373866324, 11.11904138979044991149374134566, 11.59348760461196207022916201129, 13.29981118013829432079206445953

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