Properties

Degree $2$
Conductor $128$
Sign $0.524 + 0.851i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.63 − 4.63i)3-s + (29.2 − 29.2i)5-s + 59.6·7-s + 38.0i·9-s + (18.0 + 18.0i)11-s + (−50.7 − 50.7i)13-s − 270. i·15-s − 223.·17-s + (−14.7 + 14.7i)19-s + (276. − 276. i)21-s + 739.·23-s − 1.08e3i·25-s + (551. + 551. i)27-s + (−938. − 938. i)29-s + 938. i·31-s + ⋯
L(s)  = 1  + (0.515 − 0.515i)3-s + (1.16 − 1.16i)5-s + 1.21·7-s + 0.469i·9-s + (0.149 + 0.149i)11-s + (−0.300 − 0.300i)13-s − 1.20i·15-s − 0.774·17-s + (−0.0408 + 0.0408i)19-s + (0.626 − 0.626i)21-s + 1.39·23-s − 1.72i·25-s + (0.756 + 0.756i)27-s + (−1.11 − 1.11i)29-s + 0.976i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.524 + 0.851i$
Motivic weight: \(4\)
Character: $\chi_{128} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ 0.524 + 0.851i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.45799 - 1.37308i\)
\(L(\frac12)\) \(\approx\) \(2.45799 - 1.37308i\)
\(L(3)\) 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 \( 1 + (-4.63 + 4.63i)T - 81iT^{2} \)
5 \( 1 + (-29.2 + 29.2i)T - 625iT^{2} \)
7 \( 1 - 59.6T + 2.40e3T^{2} \)
11 \( 1 + (-18.0 - 18.0i)T + 1.46e4iT^{2} \)
13 \( 1 + (50.7 + 50.7i)T + 2.85e4iT^{2} \)
17 \( 1 + 223.T + 8.35e4T^{2} \)
19 \( 1 + (14.7 - 14.7i)T - 1.30e5iT^{2} \)
23 \( 1 - 739.T + 2.79e5T^{2} \)
29 \( 1 + (938. + 938. i)T + 7.07e5iT^{2} \)
31 \( 1 - 938. iT - 9.23e5T^{2} \)
37 \( 1 + (263. - 263. i)T - 1.87e6iT^{2} \)
41 \( 1 + 248. iT - 2.82e6T^{2} \)
43 \( 1 + (1.03e3 + 1.03e3i)T + 3.41e6iT^{2} \)
47 \( 1 + 2.01e3iT - 4.87e6T^{2} \)
53 \( 1 + (833. - 833. i)T - 7.89e6iT^{2} \)
59 \( 1 + (-2.22e3 - 2.22e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-341. - 341. i)T + 1.38e7iT^{2} \)
67 \( 1 + (4.84e3 - 4.84e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 4.18e3T + 2.54e7T^{2} \)
73 \( 1 - 9.07e3iT - 2.83e7T^{2} \)
79 \( 1 + 735. iT - 3.89e7T^{2} \)
83 \( 1 + (1.44e3 - 1.44e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 5.07e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.52e3T + 8.85e7T^{2} \)
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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

−12.83949048829993660969985780790, −11.53008911107249943071360574514, −10.31193360544279554885902473847, −9.007806950036654102483244853846, −8.390863681781978392379925688746, −7.17699115752864661725672037547, −5.48028573390952417312381799387, −4.69738619504676668393352379008, −2.25822344240352098709012069140, −1.32858157613355840910566388927, 1.85767291709422381269939923441, 3.16841777510735021781998520329, 4.77915472155862230295851276338, 6.20394680919539063231416475097, 7.30446778612276831453899455142, 8.841258280701176291665306593099, 9.605224806249718561114671053583, 10.75433360874628963663451716037, 11.41140344450675096897426069243, 13.06061102993930927017448552086

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