Properties

Degree $2$
Conductor $128$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.92·3-s + 11.8·5-s − 9.85·7-s + 52.7·9-s + 39.0·11-s − 91.5·13-s + 105.·15-s − 37.1·17-s − 46.4·19-s − 87.9·21-s + 120.·23-s + 15.5·25-s + 229.·27-s + 27.2·29-s + 81.1·31-s + 348.·33-s − 116.·35-s − 10.9·37-s − 817.·39-s − 205.·41-s + 115.·43-s + 624.·45-s + 312.·47-s − 245.·49-s − 331.·51-s − 90.9·53-s + 463.·55-s + ⋯
L(s)  = 1  + 1.71·3-s + 1.06·5-s − 0.532·7-s + 1.95·9-s + 1.07·11-s − 1.95·13-s + 1.82·15-s − 0.529·17-s − 0.561·19-s − 0.914·21-s + 1.09·23-s + 0.124·25-s + 1.63·27-s + 0.174·29-s + 0.470·31-s + 1.84·33-s − 0.564·35-s − 0.0488·37-s − 3.35·39-s − 0.782·41-s + 0.409·43-s + 2.07·45-s + 0.970·47-s − 0.716·49-s − 0.910·51-s − 0.235·53-s + 1.13·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{128} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.02564\)
\(L(\frac12)\) \(\approx\) \(3.02564\)
\(L(\frac{5}{2})\) 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 \( 1 - 8.92T + 27T^{2} \)
5 \( 1 - 11.8T + 125T^{2} \)
7 \( 1 + 9.85T + 343T^{2} \)
11 \( 1 - 39.0T + 1.33e3T^{2} \)
13 \( 1 + 91.5T + 2.19e3T^{2} \)
17 \( 1 + 37.1T + 4.91e3T^{2} \)
19 \( 1 + 46.4T + 6.85e3T^{2} \)
23 \( 1 - 120.T + 1.21e4T^{2} \)
29 \( 1 - 27.2T + 2.43e4T^{2} \)
31 \( 1 - 81.1T + 2.97e4T^{2} \)
37 \( 1 + 10.9T + 5.06e4T^{2} \)
41 \( 1 + 205.T + 6.89e4T^{2} \)
43 \( 1 - 115.T + 7.95e4T^{2} \)
47 \( 1 - 312.T + 1.03e5T^{2} \)
53 \( 1 + 90.9T + 1.48e5T^{2} \)
59 \( 1 + 550.T + 2.05e5T^{2} \)
61 \( 1 + 630.T + 2.26e5T^{2} \)
67 \( 1 + 661.T + 3.00e5T^{2} \)
71 \( 1 + 494.T + 3.57e5T^{2} \)
73 \( 1 - 566.T + 3.89e5T^{2} \)
79 \( 1 + 49.4T + 4.93e5T^{2} \)
83 \( 1 - 564.T + 5.71e5T^{2} \)
89 \( 1 - 1.08e3T + 7.04e5T^{2} \)
97 \( 1 - 464.T + 9.12e5T^{2} \)
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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

−13.16154726850348948241182793717, −12.20685414735519986830343525635, −10.30198576382715341401256737536, −9.418230919452329862065684237713, −8.994498049698335360907103452765, −7.51943253153869667627807186037, −6.49360770145709687340431244092, −4.59221321474450316069971580288, −3.01160054344601032192191593912, −1.96205488991068877670545027130, 1.96205488991068877670545027130, 3.01160054344601032192191593912, 4.59221321474450316069971580288, 6.49360770145709687340431244092, 7.51943253153869667627807186037, 8.994498049698335360907103452765, 9.418230919452329862065684237713, 10.30198576382715341401256737536, 12.20685414735519986830343525635, 13.16154726850348948241182793717

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