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Properties

Label 2-160-20.7-c5-0-9
Degree $2$
Conductor $160$
Sign $0.325 - 0.945i$
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  + (−3.29 − 3.29i)3-s + (35.3 + 43.2i)5-s + (5.96 − 5.96i)7-s − 221. i·9-s + 634. i·11-s + (226. − 226. i)13-s + (25.9 − 259. i)15-s + (429. + 429. i)17-s − 540.·19-s − 39.2·21-s + (−1.34e3 − 1.34e3i)23-s + (−620. + 3.06e3i)25-s + (−1.52e3 + 1.52e3i)27-s − 1.94e3i·29-s + 8.01e3i·31-s + ⋯
L(s)  = 1  + (−0.211 − 0.211i)3-s + (0.632 + 0.774i)5-s + (0.0459 − 0.0459i)7-s − 0.910i·9-s + 1.58i·11-s + (0.371 − 0.371i)13-s + (0.0298 − 0.297i)15-s + (0.360 + 0.360i)17-s − 0.343·19-s − 0.0194·21-s + (−0.530 − 0.530i)23-s + (−0.198 + 0.980i)25-s + (−0.403 + 0.403i)27-s − 0.428i·29-s + 1.49i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.325 - 0.945i$
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: \((2,\ 160,\ (\ :5/2),\ 0.325 - 0.945i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.794745652\)
\(L(\frac12)\) \(\approx\) \(1.794745652\)
\(L(\frac{7}{2})\) 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 \( 1 + (-35.3 - 43.2i)T \)
good3 \( 1 + (3.29 + 3.29i)T + 243iT^{2} \)
7 \( 1 + (-5.96 + 5.96i)T - 1.68e4iT^{2} \)
11 \( 1 - 634. iT - 1.61e5T^{2} \)
13 \( 1 + (-226. + 226. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-429. - 429. i)T + 1.41e6iT^{2} \)
19 \( 1 + 540.T + 2.47e6T^{2} \)
23 \( 1 + (1.34e3 + 1.34e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 1.94e3iT - 2.05e7T^{2} \)
31 \( 1 - 8.01e3iT - 2.86e7T^{2} \)
37 \( 1 + (-1.10e4 - 1.10e4i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.30e4T + 1.15e8T^{2} \)
43 \( 1 + (-1.05e4 - 1.05e4i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.44e4 - 1.44e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (6.05e3 - 6.05e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 405.T + 7.14e8T^{2} \)
61 \( 1 - 1.08e4T + 8.44e8T^{2} \)
67 \( 1 + (-8.88e3 + 8.88e3i)T - 1.35e9iT^{2} \)
71 \( 1 - 1.24e4iT - 1.80e9T^{2} \)
73 \( 1 + (-1.70e4 + 1.70e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 6.90e4T + 3.07e9T^{2} \)
83 \( 1 + (-4.06e4 - 4.06e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 6.47e4iT - 5.58e9T^{2} \)
97 \( 1 + (-3.53e4 - 3.53e4i)T + 8.58e9iT^{2} \)
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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

−12.34047031608549199396455412179, −11.12192831608446796511627843396, −10.10060823531389780108207916035, −9.394380735517474330697691599464, −7.85135019048459203272688477349, −6.72510983930936088929490866667, −5.98393764789979646118169797105, −4.39164640858215376865994914154, −2.85362722903485081013931113636, −1.39635354192445867532973689711, 0.64286379314842526860615844438, 2.21795646826953005670105699888, 4.00103076860696314868052010075, 5.37851714184594652890602726554, 6.04681884086828213267459727603, 7.80023854173302845320412959940, 8.731055429916320292733566106898, 9.710165187779310554327563139383, 10.87512474817060654920470320231, 11.61595589475541924762159444423

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