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Properties

Label 2-160-20.7-c5-0-17
Degree $2$
Conductor $160$
Sign $0.642 - 0.766i$
Analytic cond. $25.6614$
Root an. cond. $5.06570$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16.6 + 16.6i)3-s + (53.0 − 17.5i)5-s + (76.6 − 76.6i)7-s + 310. i·9-s + 711. i·11-s + (467. − 467. i)13-s + (1.17e3 + 590. i)15-s + (−836. − 836. i)17-s + 2.32e3·19-s + 2.55e3·21-s + (−1.73e3 − 1.73e3i)23-s + (2.50e3 − 1.86e3i)25-s + (−1.12e3 + 1.12e3i)27-s + 4.87e3i·29-s + 3.48e3i·31-s + ⋯
L(s)  = 1  + (1.06 + 1.06i)3-s + (0.949 − 0.314i)5-s + (0.591 − 0.591i)7-s + 1.27i·9-s + 1.77i·11-s + (0.766 − 0.766i)13-s + (1.34 + 0.678i)15-s + (−0.702 − 0.702i)17-s + 1.47·19-s + 1.26·21-s + (−0.683 − 0.683i)23-s + (0.802 − 0.596i)25-s + (−0.298 + 0.298i)27-s + 1.07i·29-s + 0.652i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.672030869\)
\(L(\frac12)\) \(\approx\) \(3.672030869\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-53.0 + 17.5i)T \)
good3 \( 1 + (-16.6 - 16.6i)T + 243iT^{2} \)
7 \( 1 + (-76.6 + 76.6i)T - 1.68e4iT^{2} \)
11 \( 1 - 711. iT - 1.61e5T^{2} \)
13 \( 1 + (-467. + 467. i)T - 3.71e5iT^{2} \)
17 \( 1 + (836. + 836. i)T + 1.41e6iT^{2} \)
19 \( 1 - 2.32e3T + 2.47e6T^{2} \)
23 \( 1 + (1.73e3 + 1.73e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 4.87e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.48e3iT - 2.86e7T^{2} \)
37 \( 1 + (-4.90e3 - 4.90e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.89e4T + 1.15e8T^{2} \)
43 \( 1 + (737. + 737. i)T + 1.47e8iT^{2} \)
47 \( 1 + (-9.02e3 + 9.02e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (6.48e3 - 6.48e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 7.29e3T + 7.14e8T^{2} \)
61 \( 1 + 2.72e4T + 8.44e8T^{2} \)
67 \( 1 + (2.35e4 - 2.35e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 1.00e3iT - 1.80e9T^{2} \)
73 \( 1 + (-7.72e3 + 7.72e3i)T - 2.07e9iT^{2} \)
79 \( 1 - 7.27e4T + 3.07e9T^{2} \)
83 \( 1 + (5.19e4 + 5.19e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 6.16e4iT - 5.58e9T^{2} \)
97 \( 1 + (6.54e4 + 6.54e4i)T + 8.58e9iT^{2} \)
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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

−12.22386125203977805625100470667, −10.64462559933171536914264895734, −9.977873928366946647383786431580, −9.236263192578816655853850958253, −8.230316335359775110946440546999, −7.00438103109033849318277540573, −5.16345184114180778763677969774, −4.39073917382511909462508105025, −2.93371364103482511303464513516, −1.54394666542485852482667727548, 1.29097999647272897830669660417, 2.26623114307531245400140632273, 3.46945810394791696528885460058, 5.67260502930474100631298803270, 6.47216307575897513679471436024, 7.86990195668916758140970709893, 8.638299209899851737113645840817, 9.442738151600591407491786246232, 11.02480204979143575399524027655, 11.85374318192376351119683171987

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