Properties

Degree $2$
Conductor $128$
Sign $0.302 + 0.953i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.143 − 0.0595i)3-s + (−0.767 − 1.85i)5-s + (5.47 − 5.47i)7-s + (−19.0 − 19.0i)9-s + (36.9 − 15.2i)11-s + (4.49 − 10.8i)13-s + 0.312i·15-s − 53.8i·17-s + (31.9 − 77.2i)19-s + (−1.11 + 0.461i)21-s + (−50.0 − 50.0i)23-s + (85.5 − 85.5i)25-s + (3.21 + 7.75i)27-s + (156. + 64.6i)29-s − 207.·31-s + ⋯
L(s)  = 1  + (−0.0276 − 0.0114i)3-s + (−0.0686 − 0.165i)5-s + (0.295 − 0.295i)7-s + (−0.706 − 0.706i)9-s + (1.01 − 0.419i)11-s + (0.0959 − 0.231i)13-s + 0.00537i·15-s − 0.768i·17-s + (0.386 − 0.932i)19-s + (−0.0115 + 0.00479i)21-s + (−0.454 − 0.454i)23-s + (0.684 − 0.684i)25-s + (0.0229 + 0.0553i)27-s + (0.999 + 0.414i)29-s − 1.20·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.302 + 0.953i$
Motivic weight: \(3\)
Character: $\chi_{128} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.302 + 0.953i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.19362 - 0.873232i\)
\(L(\frac12)\) \(\approx\) \(1.19362 - 0.873232i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.143 + 0.0595i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (0.767 + 1.85i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (-5.47 + 5.47i)T - 343iT^{2} \)
11 \( 1 + (-36.9 + 15.2i)T + (941. - 941. i)T^{2} \)
13 \( 1 + (-4.49 + 10.8i)T + (-1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 53.8iT - 4.91e3T^{2} \)
19 \( 1 + (-31.9 + 77.2i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (50.0 + 50.0i)T + 1.21e4iT^{2} \)
29 \( 1 + (-156. - 64.6i)T + (1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 207.T + 2.97e4T^{2} \)
37 \( 1 + (-73.7 - 177. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (293. + 293. i)T + 6.89e4iT^{2} \)
43 \( 1 + (-342. + 141. i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 510. iT - 1.03e5T^{2} \)
53 \( 1 + (590. - 244. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-257. - 622. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-424. - 175. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (482. + 199. i)T + (2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-199. + 199. i)T - 3.57e5iT^{2} \)
73 \( 1 + (127. + 127. i)T + 3.89e5iT^{2} \)
79 \( 1 - 237. iT - 4.93e5T^{2} \)
83 \( 1 + (9.62 - 23.2i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-329. + 329. i)T - 7.04e5iT^{2} \)
97 \( 1 - 776.T + 9.12e5T^{2} \)
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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

−12.49289359993463019400146371166, −11.67029771850641378672329123737, −10.75084086268343680663116463708, −9.305668742598941538475470402388, −8.580401201971787102731055817864, −7.13056926280828933939065693631, −6.01167194207304173480498577038, −4.54810935810526307140963386252, −3.05978855978228149738122276952, −0.813770039235749775532379016527, 1.84097788252715392389411852054, 3.66864857316653882903222476027, 5.19809201267988363114521859476, 6.41279722163412383491020435162, 7.78562281902361963148722496309, 8.781339575481765041819544611378, 9.967585536769720645186647041856, 11.16657630284919189258175464536, 11.90498298658674094309012003467, 13.04901582271185914741287504427

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