Properties

Degree $2$
Conductor $128$
Sign $-0.901 - 0.432i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.42 + 9.42i)3-s + (2.84 + 2.84i)5-s − 76.7·7-s + 96.6i·9-s + (−121. + 121. i)11-s + (−27.1 + 27.1i)13-s + 53.6i·15-s − 88.0·17-s + (261. + 261. i)19-s + (−723. − 723. i)21-s + 93.4·23-s − 608. i·25-s + (−147. + 147. i)27-s + (−272. + 272. i)29-s + 1.23e3i·31-s + ⋯
L(s)  = 1  + (1.04 + 1.04i)3-s + (0.113 + 0.113i)5-s − 1.56·7-s + 1.19i·9-s + (−1.00 + 1.00i)11-s + (−0.160 + 0.160i)13-s + 0.238i·15-s − 0.304·17-s + (0.723 + 0.723i)19-s + (−1.64 − 1.64i)21-s + 0.176·23-s − 0.974i·25-s + (−0.202 + 0.202i)27-s + (−0.324 + 0.324i)29-s + 1.28i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.901 - 0.432i$
Motivic weight: \(4\)
Character: $\chi_{128} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ -0.901 - 0.432i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.318121 + 1.39917i\)
\(L(\frac12)\) \(\approx\) \(0.318121 + 1.39917i\)
\(L(3)\) 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 \( 1 + (-9.42 - 9.42i)T + 81iT^{2} \)
5 \( 1 + (-2.84 - 2.84i)T + 625iT^{2} \)
7 \( 1 + 76.7T + 2.40e3T^{2} \)
11 \( 1 + (121. - 121. i)T - 1.46e4iT^{2} \)
13 \( 1 + (27.1 - 27.1i)T - 2.85e4iT^{2} \)
17 \( 1 + 88.0T + 8.35e4T^{2} \)
19 \( 1 + (-261. - 261. i)T + 1.30e5iT^{2} \)
23 \( 1 - 93.4T + 2.79e5T^{2} \)
29 \( 1 + (272. - 272. i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.23e3iT - 9.23e5T^{2} \)
37 \( 1 + (-1.04e3 - 1.04e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 915. iT - 2.82e6T^{2} \)
43 \( 1 + (1.11e3 - 1.11e3i)T - 3.41e6iT^{2} \)
47 \( 1 - 1.72e3iT - 4.87e6T^{2} \)
53 \( 1 + (734. + 734. i)T + 7.89e6iT^{2} \)
59 \( 1 + (-1.20e3 + 1.20e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (580. - 580. i)T - 1.38e7iT^{2} \)
67 \( 1 + (-1.48e3 - 1.48e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 5.57e3T + 2.54e7T^{2} \)
73 \( 1 + 6.61e3iT - 2.83e7T^{2} \)
79 \( 1 + 5.39e3iT - 3.89e7T^{2} \)
83 \( 1 + (-2.55e3 - 2.55e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.09e4iT - 6.27e7T^{2} \)
97 \( 1 - 4.71e3T + 8.85e7T^{2} \)
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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

−13.17064294296058988475298337345, −12.28657783957946566138060878529, −10.45691216720593058220627134875, −9.886626429296688268588019122858, −9.177789083426086248488259337745, −7.88400501498535624530932977506, −6.55868353868128556188255497351, −4.89970221335342150662927830175, −3.54857013283526561380345559722, −2.58611342607392751001721279189, 0.49515916124463624227045772223, 2.50683006512398404937413056931, 3.38272418526351657910539209363, 5.67840321224544808309786086394, 6.88981664391627716681001939942, 7.82358488798082605361322833597, 8.943448934112665045065182281288, 9.803105764548043818247359435657, 11.26376606583655674894016617184, 12.74489197760186211386319934066

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