Properties

Degree $2$
Conductor $128$
Sign $-0.959 + 0.281i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.374 + 0.155i)3-s + (−7.60 + 3.15i)5-s + (−6.84 − 6.84i)7-s + (−6.24 − 6.24i)9-s + (2.23 − 0.927i)11-s + (1.40 + 0.583i)13-s − 3.34·15-s + 2.67i·17-s + (−5.38 + 13.0i)19-s + (−1.50 − 3.62i)21-s + (−18.8 + 18.8i)23-s + (30.2 − 30.2i)25-s + (−2.77 − 6.68i)27-s + (−10.0 + 24.2i)29-s − 47.5i·31-s + ⋯
L(s)  = 1  + (0.124 + 0.0517i)3-s + (−1.52 + 0.630i)5-s + (−0.977 − 0.977i)7-s + (−0.694 − 0.694i)9-s + (0.203 − 0.0842i)11-s + (0.108 + 0.0449i)13-s − 0.222·15-s + 0.157i·17-s + (−0.283 + 0.684i)19-s + (−0.0715 − 0.172i)21-s + (−0.819 + 0.819i)23-s + (1.21 − 1.21i)25-s + (−0.102 − 0.247i)27-s + (−0.345 + 0.834i)29-s − 1.53i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.959 + 0.281i$
Motivic weight: \(2\)
Character: $\chi_{128} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ -0.959 + 0.281i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0206305 - 0.143455i\)
\(L(\frac12)\) \(\approx\) \(0.0206305 - 0.143455i\)
\(L(2)\) 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 \( 1 + (-0.374 - 0.155i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (7.60 - 3.15i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (6.84 + 6.84i)T + 49iT^{2} \)
11 \( 1 + (-2.23 + 0.927i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-1.40 - 0.583i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 2.67iT - 289T^{2} \)
19 \( 1 + (5.38 - 13.0i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (18.8 - 18.8i)T - 529iT^{2} \)
29 \( 1 + (10.0 - 24.2i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 47.5iT - 961T^{2} \)
37 \( 1 + (28.2 - 11.7i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-6.93 - 6.93i)T + 1.68e3iT^{2} \)
43 \( 1 + (8.48 - 3.51i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 67.0T + 2.20e3T^{2} \)
53 \( 1 + (10.5 + 25.3i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (27.9 + 67.4i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-31.5 + 76.2i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (90.1 + 37.3i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (1.98 + 1.98i)T + 5.04e3iT^{2} \)
73 \( 1 + (55.5 + 55.5i)T + 5.32e3iT^{2} \)
79 \( 1 + 10.9T + 6.24e3T^{2} \)
83 \( 1 + (34.1 - 82.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (16.1 - 16.1i)T - 7.92e3iT^{2} \)
97 \( 1 + 62.6T + 9.40e3T^{2} \)
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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

−12.47496707941512667390425192244, −11.60595080882033676366457892007, −10.69464934610794993866882210631, −9.541845196148372201603370841930, −8.173398909362859642221461073204, −7.22384055230301137141017319519, −6.18432985541441755029254882311, −3.94319465587937226550231336640, −3.38523456439259574043439321676, −0.093822894798920977570877457110, 2.83489168333584440339155801516, 4.30239429977187771914618904374, 5.71065837611223431535061955831, 7.21410982503443069307334970372, 8.452498548286397817312169297105, 8.987784608104906499460673138920, 10.60649812239359359573230007333, 11.83713852382131144339096832474, 12.30863431494304449012478459088, 13.39383571638972828789571472962

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