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Properties

Label 2-2e7-4.3-c4-0-5
Degree $2$
Conductor $128$
Sign $-i$
Analytic cond. $13.2313$
Root an. cond. $3.63749$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.29i·3-s + 22.7·5-s + 51.5i·7-s + 41.3·9-s − 50.8i·11-s − 160.·13-s + 143. i·15-s + 416.·17-s + 515. i·19-s − 324.·21-s + 15.8i·23-s − 107.·25-s + 770. i·27-s − 979.·29-s + 1.90e3i·31-s + ⋯
L(s)  = 1  + 0.699i·3-s + 0.909·5-s + 1.05i·7-s + 0.510·9-s − 0.419i·11-s − 0.949·13-s + 0.636i·15-s + 1.44·17-s + 1.42i·19-s − 0.735·21-s + 0.0299i·23-s − 0.172·25-s + 1.05i·27-s − 1.16·29-s + 1.98i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-i$
Analytic conductor: \(13.2313\)
Root analytic conductor: \(3.63749\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ -i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.43273 + 1.43273i\)
\(L(\frac12)\) \(\approx\) \(1.43273 + 1.43273i\)
\(L(3)\) 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 \( 1 - 6.29iT - 81T^{2} \)
5 \( 1 - 22.7T + 625T^{2} \)
7 \( 1 - 51.5iT - 2.40e3T^{2} \)
11 \( 1 + 50.8iT - 1.46e4T^{2} \)
13 \( 1 + 160.T + 2.85e4T^{2} \)
17 \( 1 - 416.T + 8.35e4T^{2} \)
19 \( 1 - 515. iT - 1.30e5T^{2} \)
23 \( 1 - 15.8iT - 2.79e5T^{2} \)
29 \( 1 + 979.T + 7.07e5T^{2} \)
31 \( 1 - 1.90e3iT - 9.23e5T^{2} \)
37 \( 1 + 657.T + 1.87e6T^{2} \)
41 \( 1 - 2.83e3T + 2.82e6T^{2} \)
43 \( 1 + 2.53e3iT - 3.41e6T^{2} \)
47 \( 1 - 940. iT - 4.87e6T^{2} \)
53 \( 1 - 280.T + 7.89e6T^{2} \)
59 \( 1 + 5.03e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.88e3T + 1.38e7T^{2} \)
67 \( 1 - 1.42e3iT - 2.01e7T^{2} \)
71 \( 1 + 8.92e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.47e3T + 2.83e7T^{2} \)
79 \( 1 + 6.83e3iT - 3.89e7T^{2} \)
83 \( 1 + 4.78e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.70e3T + 6.27e7T^{2} \)
97 \( 1 - 1.89e3T + 8.85e7T^{2} \)
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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

−12.70081196304685309276031880758, −12.05220286108066056442379500471, −10.49164910662089664398923406045, −9.812426121422127736490891531075, −8.972795168255405982888437432637, −7.54753574949955762895849368249, −5.89464443366090608889927948570, −5.17389556023166717671503771732, −3.44568607053140771631002752346, −1.82166409912513387541847726951, 0.894614068115663459867021985419, 2.31975226373503970964119884052, 4.29012543299645680334869544648, 5.75203187971830556995093797452, 7.12806499164660761852150917861, 7.64083150291039022344153996380, 9.554417087592971451643914022894, 10.05288616752259809783961423542, 11.40502078759957672829725348988, 12.69420838640110451803255725833

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