Properties

Degree $2$
Conductor $128$
Sign $0.632 + 0.774i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.68 − 1.94i)3-s + (−4.51 − 1.86i)5-s + (3.85 − 3.85i)7-s + (11.8 − 11.8i)9-s + (4.56 + 1.89i)11-s + (−5.58 + 2.31i)13-s − 24.7·15-s + 25.0i·17-s + (−6.43 − 15.5i)19-s + (10.5 − 25.5i)21-s + (26.9 + 26.9i)23-s + (−0.825 − 0.825i)25-s + (15.0 − 36.2i)27-s + (−0.210 − 0.507i)29-s + 15.8i·31-s + ⋯
L(s)  = 1  + (1.56 − 0.647i)3-s + (−0.902 − 0.373i)5-s + (0.550 − 0.550i)7-s + (1.31 − 1.31i)9-s + (0.414 + 0.171i)11-s + (−0.429 + 0.177i)13-s − 1.65·15-s + 1.47i·17-s + (−0.338 − 0.817i)19-s + (0.503 − 1.21i)21-s + (1.16 + 1.16i)23-s + (−0.0330 − 0.0330i)25-s + (0.556 − 1.34i)27-s + (−0.00724 − 0.0174i)29-s + 0.510i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.632 + 0.774i$
Motivic weight: \(2\)
Character: $\chi_{128} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1),\ 0.632 + 0.774i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.84238 - 0.874041i\)
\(L(\frac12)\) \(\approx\) \(1.84238 - 0.874041i\)
\(L(2)\) 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 \( 1 + (-4.68 + 1.94i)T + (6.36 - 6.36i)T^{2} \)
5 \( 1 + (4.51 + 1.86i)T + (17.6 + 17.6i)T^{2} \)
7 \( 1 + (-3.85 + 3.85i)T - 49iT^{2} \)
11 \( 1 + (-4.56 - 1.89i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + (5.58 - 2.31i)T + (119. - 119. i)T^{2} \)
17 \( 1 - 25.0iT - 289T^{2} \)
19 \( 1 + (6.43 + 15.5i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (-26.9 - 26.9i)T + 529iT^{2} \)
29 \( 1 + (0.210 + 0.507i)T + (-594. + 594. i)T^{2} \)
31 \( 1 - 15.8iT - 961T^{2} \)
37 \( 1 + (2.18 + 0.905i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (31.1 - 31.1i)T - 1.68e3iT^{2} \)
43 \( 1 + (12.9 + 5.34i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 15.0T + 2.20e3T^{2} \)
53 \( 1 + (15.4 - 37.2i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-14.7 + 35.5i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (15.4 + 37.3i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (61.3 - 25.4i)T + (3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-51.7 + 51.7i)T - 5.04e3iT^{2} \)
73 \( 1 + (-64.9 + 64.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 38.1T + 6.24e3T^{2} \)
83 \( 1 + (15.9 + 38.5i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-23.7 - 23.7i)T + 7.92e3iT^{2} \)
97 \( 1 + 118.T + 9.40e3T^{2} \)
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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

−13.08706067667644792087561144962, −12.20314458727652743323902745726, −10.97959010146388405023142766577, −9.451558500084266455711369134011, −8.472216166390064865468292007844, −7.77646769907680788310769647393, −6.85452633284929121285278119143, −4.50748380454427188329007168033, −3.37997471603222083800801643521, −1.57739520964253332128572253784, 2.53040949784029341184739263967, 3.67739814316873781162943889190, 4.91320578595944712807716074550, 7.11482281604610966555377426271, 8.124758710052382730282867030531, 8.882200943905137627600916272761, 9.909037050492496991136128929581, 11.15554550600433587747551563816, 12.18043611976798271503865683783, 13.55381688603594931479512489498

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