Properties

Degree $2$
Conductor $128$
Sign $-0.966 + 0.258i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.54 + 5.54i)3-s + (−21.7 − 21.7i)5-s + 6.62·7-s − 19.6i·9-s + (−90.9 + 90.9i)11-s + (−221. + 221. i)13-s − 240. i·15-s − 132.·17-s + (−402. − 402. i)19-s + (36.6 + 36.6i)21-s − 27.5·23-s + 320. i·25-s + (557. − 557. i)27-s + (−174. + 174. i)29-s − 1.08e3i·31-s + ⋯
L(s)  = 1  + (0.615 + 0.615i)3-s + (−0.869 − 0.869i)5-s + 0.135·7-s − 0.242i·9-s + (−0.752 + 0.752i)11-s + (−1.31 + 1.31i)13-s − 1.07i·15-s − 0.458·17-s + (−1.11 − 1.11i)19-s + (0.0831 + 0.0831i)21-s − 0.0519·23-s + 0.512i·25-s + (0.764 − 0.764i)27-s + (−0.207 + 0.207i)29-s − 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.966 + 0.258i$
Motivic weight: \(4\)
Character: $\chi_{128} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ -0.966 + 0.258i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0101240 - 0.0769992i\)
\(L(\frac12)\) \(\approx\) \(0.0101240 - 0.0769992i\)
\(L(3)\) 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 \( 1 + (-5.54 - 5.54i)T + 81iT^{2} \)
5 \( 1 + (21.7 + 21.7i)T + 625iT^{2} \)
7 \( 1 - 6.62T + 2.40e3T^{2} \)
11 \( 1 + (90.9 - 90.9i)T - 1.46e4iT^{2} \)
13 \( 1 + (221. - 221. i)T - 2.85e4iT^{2} \)
17 \( 1 + 132.T + 8.35e4T^{2} \)
19 \( 1 + (402. + 402. i)T + 1.30e5iT^{2} \)
23 \( 1 + 27.5T + 2.79e5T^{2} \)
29 \( 1 + (174. - 174. i)T - 7.07e5iT^{2} \)
31 \( 1 + 1.08e3iT - 9.23e5T^{2} \)
37 \( 1 + (553. + 553. i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.80e3iT - 2.82e6T^{2} \)
43 \( 1 + (-17.8 + 17.8i)T - 3.41e6iT^{2} \)
47 \( 1 - 2.26e3iT - 4.87e6T^{2} \)
53 \( 1 + (-822. - 822. i)T + 7.89e6iT^{2} \)
59 \( 1 + (972. - 972. i)T - 1.21e7iT^{2} \)
61 \( 1 + (-2.05e3 + 2.05e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (-4.61e3 - 4.61e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 3.10e3T + 2.54e7T^{2} \)
73 \( 1 + 723. iT - 2.83e7T^{2} \)
79 \( 1 + 3.41e3iT - 3.89e7T^{2} \)
83 \( 1 + (161. + 161. i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.46e3iT - 6.27e7T^{2} \)
97 \( 1 + 8.26e3T + 8.85e7T^{2} \)
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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

−12.23164819215256632476865749114, −11.22400455264721240669725572400, −9.769386343181256234620926553847, −9.054469565351955462983637984636, −8.037987933457696567090552907535, −6.85298120680978566881466648711, −4.73165082340308640942486802189, −4.25213017562983597458560496115, −2.39281634571362825484142718452, −0.02770355095177397624396912803, 2.34458125383439833380869397453, 3.42798132716934368461864979620, 5.24398408801333593246759118683, 6.89416460520642841276622100713, 7.85181044226004210971088427528, 8.378363941741067394634602669489, 10.27884980989936724283420397785, 10.90972556150189613980629029870, 12.25109891667794787951933865277, 13.06948646855260768563726683057

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