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Properties

Label 2-2e7-16.3-c4-0-2
Degree $2$
Conductor $128$
Sign $0.159 - 0.987i$
Analytic cond. $13.2313$
Root an. cond. $3.63749$
Motivic weight $4$
Arithmetic yes
Rational no
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

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

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.159 - 0.987i$
Analytic conductor: \(13.2313\)
Root analytic conductor: \(3.63749\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ 0.159 - 0.987i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.07797 + 0.917720i\)
\(L(\frac12)\) \(\approx\) \(1.07797 + 0.917720i\)
\(L(3)\) 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

−12.66837503434087980914857426386, −12.20694904107290354611826721280, −10.50662962002507821104072413380, −9.887321457442243497812202339080, −8.687922453944076037752763023982, −7.43840677233148158108339323358, −6.20984337279012424748415121728, −4.97597790436380508822774763733, −3.40323639575062899017620124526, −1.63731207090663851589154453901, 0.60772629385053788671566438609, 2.81993404040356322736284380478, 4.04336149855134639815764966163, 6.09416559391254557045724245125, 6.53403356993108786325801000806, 8.185918383761098180990055559504, 9.485285058181252795670799645743, 10.05567696698630676013465925138, 11.52748356943772147916585373100, 12.36011997182082869616822787119

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