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Properties

Label 2-160-8.5-c7-0-1
Degree $2$
Conductor $160$
Sign $-0.176 + 0.984i$
Analytic cond. $49.9816$
Root an. cond. $7.06976$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 69.9i·3-s + 125i·5-s + 814.·7-s − 2.70e3·9-s − 1.99e3i·11-s + 2.04e3i·13-s − 8.74e3·15-s − 2.25e4·17-s − 3.85e4i·19-s + 5.69e4i·21-s − 1.06e5·23-s − 1.56e4·25-s − 3.63e4i·27-s + 1.06e5i·29-s − 1.99e5·31-s + ⋯
L(s)  = 1  + 1.49i·3-s + 0.447i·5-s + 0.897·7-s − 1.23·9-s − 0.451i·11-s + 0.257i·13-s − 0.668·15-s − 1.11·17-s − 1.28i·19-s + 1.34i·21-s − 1.82·23-s − 0.199·25-s − 0.355i·27-s + 0.812i·29-s − 1.20·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.176 + 0.984i$
Analytic conductor: \(49.9816\)
Root analytic conductor: \(7.06976\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :7/2),\ -0.176 + 0.984i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.1314284490\)
\(L(\frac12)\) \(\approx\) \(0.1314284490\)
\(L(\frac{9}{2})\) 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 \( 1 - 125iT \)
good3 \( 1 - 69.9iT - 2.18e3T^{2} \)
7 \( 1 - 814.T + 8.23e5T^{2} \)
11 \( 1 + 1.99e3iT - 1.94e7T^{2} \)
13 \( 1 - 2.04e3iT - 6.27e7T^{2} \)
17 \( 1 + 2.25e4T + 4.10e8T^{2} \)
19 \( 1 + 3.85e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.06e5T + 3.40e9T^{2} \)
29 \( 1 - 1.06e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.99e5T + 2.75e10T^{2} \)
37 \( 1 + 8.81e4iT - 9.49e10T^{2} \)
41 \( 1 + 5.69e5T + 1.94e11T^{2} \)
43 \( 1 + 8.91e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.03e6T + 5.06e11T^{2} \)
53 \( 1 + 1.69e5iT - 1.17e12T^{2} \)
59 \( 1 - 1.34e6iT - 2.48e12T^{2} \)
61 \( 1 - 2.04e6iT - 3.14e12T^{2} \)
67 \( 1 + 2.79e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.12e6T + 9.09e12T^{2} \)
73 \( 1 - 1.35e6T + 1.10e13T^{2} \)
79 \( 1 + 7.48e5T + 1.92e13T^{2} \)
83 \( 1 + 8.31e6iT - 2.71e13T^{2} \)
89 \( 1 + 4.26e6T + 4.42e13T^{2} \)
97 \( 1 + 1.04e7T + 8.07e13T^{2} \)
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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

−11.89499195914228471686924149762, −10.98591887856827582438020708918, −10.49914913904289758951285394198, −9.259651313163710249549726794278, −8.524069654332202588636834548878, −7.06209766127881660352173640228, −5.58357013348470307278229741070, −4.56264682563487646337035471493, −3.64688738441377108741159437513, −2.14623951246357107290159086765, 0.03241057982183606904235369624, 1.47821896823022596700039418103, 2.14459745804065805660738415999, 4.17623571650753219920008702975, 5.59800940093050544078571944568, 6.65741345015562020267137878632, 7.88584744381632067515958556779, 8.254450121015228706123762359339, 9.766671610359108074831153479284, 11.16772074028891839995587754732

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