Properties

Degree $2$
Conductor $128$
Sign $0.987 - 0.159i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0461 − 0.0461i)3-s + (8.04 + 8.04i)5-s + 49.8·7-s − 80.9i·9-s + (−84.2 + 84.2i)11-s + (−19.4 + 19.4i)13-s − 0.743i·15-s + 437.·17-s + (349. + 349. i)19-s + (−2.30 − 2.30i)21-s + 404.·23-s − 495. i·25-s + (−7.48 + 7.48i)27-s + (1.03e3 − 1.03e3i)29-s + 1.50e3i·31-s + ⋯
L(s)  = 1  + (−0.00513 − 0.00513i)3-s + (0.321 + 0.321i)5-s + 1.01·7-s − 0.999i·9-s + (−0.696 + 0.696i)11-s + (−0.115 + 0.115i)13-s − 0.00330i·15-s + 1.51·17-s + (0.966 + 0.966i)19-s + (−0.00522 − 0.00522i)21-s + 0.765·23-s − 0.792i·25-s + (−0.0102 + 0.0102i)27-s + (1.22 − 1.22i)29-s + 1.56i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.987 - 0.159i$
Motivic weight: \(4\)
Character: $\chi_{128} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ 0.987 - 0.159i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.11284 + 0.169657i\)
\(L(\frac12)\) \(\approx\) \(2.11284 + 0.169657i\)
\(L(3)\) 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 \( 1 + (0.0461 + 0.0461i)T + 81iT^{2} \)
5 \( 1 + (-8.04 - 8.04i)T + 625iT^{2} \)
7 \( 1 - 49.8T + 2.40e3T^{2} \)
11 \( 1 + (84.2 - 84.2i)T - 1.46e4iT^{2} \)
13 \( 1 + (19.4 - 19.4i)T - 2.85e4iT^{2} \)
17 \( 1 - 437.T + 8.35e4T^{2} \)
19 \( 1 + (-349. - 349. i)T + 1.30e5iT^{2} \)
23 \( 1 - 404.T + 2.79e5T^{2} \)
29 \( 1 + (-1.03e3 + 1.03e3i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.50e3iT - 9.23e5T^{2} \)
37 \( 1 + (-434. - 434. i)T + 1.87e6iT^{2} \)
41 \( 1 - 696. iT - 2.82e6T^{2} \)
43 \( 1 + (-917. + 917. i)T - 3.41e6iT^{2} \)
47 \( 1 + 111. iT - 4.87e6T^{2} \)
53 \( 1 + (1.04e3 + 1.04e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (1.71e3 - 1.71e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (3.71e3 - 3.71e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (1.85e3 + 1.85e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 1.16e3T + 2.54e7T^{2} \)
73 \( 1 + 905. iT - 2.83e7T^{2} \)
79 \( 1 + 5.86e3iT - 3.89e7T^{2} \)
83 \( 1 + (7.56e3 + 7.56e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 6.43e3iT - 6.27e7T^{2} \)
97 \( 1 + 413.T + 8.85e7T^{2} \)
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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

−12.38946556156505241374282096604, −11.85706433686970837480288517119, −10.43076766621148181909103395801, −9.721272128490804764584966518381, −8.283671896958569177167182144913, −7.31286073968138732839869564476, −5.92868467045171099668012822826, −4.71300689913107941515729560363, −3.03872290986641721271293481373, −1.27753374680287605346283838831, 1.18465291604728923377790811228, 2.87593749472730154089441073339, 4.91033338934636636078380116103, 5.52867079048743171213983592423, 7.46714368061338623250908225487, 8.185584081227561186744959591641, 9.451909428966505946738451796350, 10.71528538349516247518241429795, 11.39715568862577033767193079345, 12.72383961990779625902890128968

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