Properties

Degree $2$
Conductor $128$
Sign $-0.238 + 0.971i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − 4.18i)3-s + (−1.85 − 4.48i)5-s + (5.27 + 5.27i)7-s + (−8.12 − 8.12i)9-s + (−6.20 − 14.9i)11-s + (−4.22 + 10.2i)13-s − 21.9·15-s − 2.84i·17-s + (12.4 + 5.14i)19-s + (31.2 − 12.9i)21-s + (1.43 − 1.43i)23-s + (0.999 − 0.999i)25-s + (−10.4 + 4.30i)27-s + (36.9 + 15.3i)29-s + 4.73i·31-s + ⋯
L(s)  = 1  + (0.577 − 1.39i)3-s + (−0.371 − 0.897i)5-s + (0.753 + 0.753i)7-s + (−0.902 − 0.902i)9-s + (−0.563 − 1.36i)11-s + (−0.325 + 0.784i)13-s − 1.46·15-s − 0.167i·17-s + (0.654 + 0.270i)19-s + (1.48 − 0.615i)21-s + (0.0625 − 0.0625i)23-s + (0.0399 − 0.0399i)25-s + (−0.385 + 0.159i)27-s + (1.27 + 0.527i)29-s + 0.152i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.238 + 0.971i$
Motivic weight: \(2\)
Character: $\chi_{128} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ -0.238 + 0.971i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.979863 - 1.24956i\)
\(L(\frac12)\) \(\approx\) \(0.979863 - 1.24956i\)
\(L(2)\) 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 \( 1 + (-1.73 + 4.18i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (1.85 + 4.48i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (-5.27 - 5.27i)T + 49iT^{2} \)
11 \( 1 + (6.20 + 14.9i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (4.22 - 10.2i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 2.84iT - 289T^{2} \)
19 \( 1 + (-12.4 - 5.14i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-1.43 + 1.43i)T - 529iT^{2} \)
29 \( 1 + (-36.9 - 15.3i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 4.73iT - 961T^{2} \)
37 \( 1 + (6.68 + 16.1i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-40.4 - 40.4i)T + 1.68e3iT^{2} \)
43 \( 1 + (-24.5 - 59.1i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 16.5T + 2.20e3T^{2} \)
53 \( 1 + (46.9 - 19.4i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (50.0 - 20.7i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (54.3 + 22.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-25.5 + 61.5i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (7.12 + 7.12i)T + 5.04e3iT^{2} \)
73 \( 1 + (-55.3 - 55.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 11.0T + 6.24e3T^{2} \)
83 \( 1 + (-29.9 - 12.4i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-16.7 + 16.7i)T - 7.92e3iT^{2} \)
97 \( 1 + 67.8T + 9.40e3T^{2} \)
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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

−12.70162847759160039948426581337, −12.09092856869182609565068616503, −11.13261843417621113346223917486, −9.137862137605189886549924321921, −8.345481485135774414341062514473, −7.71910001538980027925895105227, −6.23515069393983299527392438508, −4.88539975766993845461208691805, −2.76142290333633326708123649921, −1.17047685750899364341789108524, 2.79856222501199387594671159639, 4.11855271416064309813104810501, 5.06932062708182164429080425182, 7.18067240850185457065747997142, 8.003656304371500005242311545888, 9.474569263795208951689745746869, 10.42067303162297565424612140980, 10.82424585986205394224951007461, 12.25920664182862747852609615283, 13.79064989262619979543585014748

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