Properties

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

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 6·5-s − 20·7-s − 23·9-s − 14·11-s + 54·13-s − 12·15-s − 66·17-s − 162·19-s + 40·21-s − 172·23-s − 89·25-s + 100·27-s − 2·29-s + 128·31-s + 28·33-s − 120·35-s + 158·37-s − 108·39-s + 202·41-s + 298·43-s − 138·45-s + 408·47-s + 57·49-s + 132·51-s − 690·53-s − 84·55-s + ⋯
L(s)  = 1  − 0.384·3-s + 0.536·5-s − 1.07·7-s − 0.851·9-s − 0.383·11-s + 1.15·13-s − 0.206·15-s − 0.941·17-s − 1.95·19-s + 0.415·21-s − 1.55·23-s − 0.711·25-s + 0.712·27-s − 0.0128·29-s + 0.741·31-s + 0.147·33-s − 0.579·35-s + 0.702·37-s − 0.443·39-s + 0.769·41-s + 1.05·43-s − 0.457·45-s + 1.26·47-s + 0.166·49-s + 0.362·51-s − 1.78·53-s − 0.205·55-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: \(1\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 2 T + p^{3} T^{2} \)
5 \( 1 - 6 T + p^{3} T^{2} \)
7 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 + 14 T + p^{3} T^{2} \)
13 \( 1 - 54 T + p^{3} T^{2} \)
17 \( 1 + 66 T + p^{3} T^{2} \)
19 \( 1 + 162 T + p^{3} T^{2} \)
23 \( 1 + 172 T + p^{3} T^{2} \)
29 \( 1 + 2 T + p^{3} T^{2} \)
31 \( 1 - 128 T + p^{3} T^{2} \)
37 \( 1 - 158 T + p^{3} T^{2} \)
41 \( 1 - 202 T + p^{3} T^{2} \)
43 \( 1 - 298 T + p^{3} T^{2} \)
47 \( 1 - 408 T + p^{3} T^{2} \)
53 \( 1 + 690 T + p^{3} T^{2} \)
59 \( 1 - 322 T + p^{3} T^{2} \)
61 \( 1 + 298 T + p^{3} T^{2} \)
67 \( 1 + 202 T + p^{3} T^{2} \)
71 \( 1 - 700 T + p^{3} T^{2} \)
73 \( 1 + 418 T + p^{3} T^{2} \)
79 \( 1 + 744 T + p^{3} T^{2} \)
83 \( 1 - 678 T + p^{3} T^{2} \)
89 \( 1 + 82 T + p^{3} T^{2} \)
97 \( 1 + 1122 T + p^{3} T^{2} \)
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

−12.49250927433566144823884122458, −11.19309127772426982466805386898, −10.38419455022517089481749640489, −9.204753736933183415426676755629, −8.194807636114177184211926510410, −6.28415107340085725171821870300, −6.03729805240788441996694338466, −4.11908511555623072926490989423, −2.43405629277512544012426618019, 0, 2.43405629277512544012426618019, 4.11908511555623072926490989423, 6.03729805240788441996694338466, 6.28415107340085725171821870300, 8.194807636114177184211926510410, 9.204753736933183415426676755629, 10.38419455022517089481749640489, 11.19309127772426982466805386898, 12.49250927433566144823884122458

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