Properties

Label 2-160-8.5-c5-0-12
Degree $2$
Conductor $160$
Sign $0.664 + 0.747i$
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  + 25.0i·3-s + 25i·5-s − 103.·7-s − 384.·9-s − 740. i·11-s − 892. i·13-s − 626.·15-s + 1.13e3·17-s + 1.15e3i·19-s − 2.59e3i·21-s − 1.60e3·23-s − 625·25-s − 3.54e3i·27-s + 2.15e3i·29-s − 4.95e3·31-s + ⋯
L(s)  = 1  + 1.60i·3-s + 0.447i·5-s − 0.799·7-s − 1.58·9-s − 1.84i·11-s − 1.46i·13-s − 0.718·15-s + 0.955·17-s + 0.734i·19-s − 1.28i·21-s − 0.631·23-s − 0.200·25-s − 0.936i·27-s + 0.476i·29-s − 0.926·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.664 + 0.747i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.664 + 0.747i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7948737247\)
\(L(\frac12)\) \(\approx\) \(0.7948737247\)
\(L(\frac{7}{2})\) 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 \( 1 - 25iT \)
good3 \( 1 - 25.0iT - 243T^{2} \)
7 \( 1 + 103.T + 1.68e4T^{2} \)
11 \( 1 + 740. iT - 1.61e5T^{2} \)
13 \( 1 + 892. iT - 3.71e5T^{2} \)
17 \( 1 - 1.13e3T + 1.41e6T^{2} \)
19 \( 1 - 1.15e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.60e3T + 6.43e6T^{2} \)
29 \( 1 - 2.15e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.95e3T + 2.86e7T^{2} \)
37 \( 1 - 4.40e3iT - 6.93e7T^{2} \)
41 \( 1 - 3.78e3T + 1.15e8T^{2} \)
43 \( 1 + 1.30e4iT - 1.47e8T^{2} \)
47 \( 1 - 8.00e3T + 2.29e8T^{2} \)
53 \( 1 + 3.43e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.20e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.82e3iT - 8.44e8T^{2} \)
67 \( 1 + 5.49e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.28e4T + 1.80e9T^{2} \)
73 \( 1 + 2.08e4T + 2.07e9T^{2} \)
79 \( 1 + 3.02e4T + 3.07e9T^{2} \)
83 \( 1 + 9.39e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.17e3T + 5.58e9T^{2} \)
97 \( 1 - 7.41e4T + 8.58e9T^{2} \)
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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

−11.52162732307164379903128968669, −10.49803987293822008225888022150, −10.12421724144372631992936191065, −8.958066292785603760530704410175, −7.929791807972240720070347364579, −6.05066734353847133311865873473, −5.37635300846805930602325236200, −3.50414458465136102847751779713, −3.27140124774868378115605725140, −0.26830791471723154911605044202, 1.35836837350507643172736036392, 2.38961404938964920731693102829, 4.34889239831630825474061988152, 5.97225907661297231886929476819, 7.02347936398138485380125186956, 7.56334117771053188897486134911, 9.024702440665294873365977879974, 9.876097290830951109107246269058, 11.59970055168734669492484450112, 12.37031984570382866855603181347

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