Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.79·3-s − 8i·5-s + 78.3i·7-s + 14.9·9-s + 107.·11-s − 216i·13-s + 78.3i·15-s − 162·17-s − 440.·19-s − 767. i·21-s − 705. i·23-s + 561·25-s + 646.·27-s − 1.30e3i·29-s − 627. i·31-s + ⋯
L(s)  = 1  − 1.08·3-s − 0.320i·5-s + 1.59i·7-s + 0.185·9-s + 0.890·11-s − 1.27i·13-s + 0.348i·15-s − 0.560·17-s − 1.22·19-s − 1.74i·21-s − 1.33i·23-s + 0.897·25-s + 0.887·27-s − 1.55i·29-s − 0.652i·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: $i$
Motivic weight: \(4\)
Character: $\chi_{128} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.510482 - 0.510482i\)
\(L(\frac12)\) \(\approx\) \(0.510482 - 0.510482i\)
\(L(3)\) 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 \( 1 + 9.79T + 81T^{2} \)
5 \( 1 + 8iT - 625T^{2} \)
7 \( 1 - 78.3iT - 2.40e3T^{2} \)
11 \( 1 - 107.T + 1.46e4T^{2} \)
13 \( 1 + 216iT - 2.85e4T^{2} \)
17 \( 1 + 162T + 8.35e4T^{2} \)
19 \( 1 + 440.T + 1.30e5T^{2} \)
23 \( 1 + 705. iT - 2.79e5T^{2} \)
29 \( 1 + 1.30e3iT - 7.07e5T^{2} \)
31 \( 1 + 627. iT - 9.23e5T^{2} \)
37 \( 1 + 1.51e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.89e3T + 2.82e6T^{2} \)
43 \( 1 + 2.90e3T + 3.41e6T^{2} \)
47 \( 1 - 1.41e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.97e3iT - 7.89e6T^{2} \)
59 \( 1 - 2.26e3T + 1.21e7T^{2} \)
61 \( 1 - 2.37e3iT - 1.38e7T^{2} \)
67 \( 1 - 1.67e3T + 2.01e7T^{2} \)
71 \( 1 + 7.75e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.75e3T + 2.83e7T^{2} \)
79 \( 1 + 7.99e3iT - 3.89e7T^{2} \)
83 \( 1 + 9.33e3T + 4.74e7T^{2} \)
89 \( 1 - 2.43e3T + 6.27e7T^{2} \)
97 \( 1 - 7.45e3T + 8.85e7T^{2} \)
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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

−12.33493692089301996841171610895, −11.48275337740360618168539128718, −10.53642352789748138281131459115, −9.102775311454724622978638898187, −8.303002650445643601614777583685, −6.38381933565016609506369513656, −5.78292837073453647806220662514, −4.58293645102061420242842711241, −2.48206631853757054646724900749, −0.37982586459999192134855018660, 1.29696649093180843465772607085, 3.77985337688437496605372513292, 4.88406142502632611400847446449, 6.66415729030919833489966812178, 6.87441255364502176155336622787, 8.713018086859630720791893717218, 10.07691589561799467437696870328, 11.03611390781707402561128709377, 11.53364312666927777070427498948, 12.82495577795500070546908824875

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