Properties

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

Origins

Related objects

Downloads

Learn more about

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: $\chi_{128} (95, \cdot )$
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 \( 1 - 8.53e3T + 8.85e7T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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