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Properties

Label 2-160-40.29-c5-0-17
Degree $2$
Conductor $160$
Sign $-0.206 + 0.978i$
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  + 1.29·3-s + (−51.3 + 21.9i)5-s + 170. i·7-s − 241.·9-s − 39.8i·11-s + 537.·13-s + (−66.7 + 28.5i)15-s − 1.35e3i·17-s − 1.03e3i·19-s + 221. i·21-s + 1.61e3i·23-s + (2.15e3 − 2.26e3i)25-s − 628.·27-s − 5.91e3i·29-s − 9.61e3·31-s + ⋯
L(s)  = 1  + 0.0832·3-s + (−0.919 + 0.393i)5-s + 1.31i·7-s − 0.993·9-s − 0.0992i·11-s + 0.881·13-s + (−0.0765 + 0.0327i)15-s − 1.13i·17-s − 0.660i·19-s + 0.109i·21-s + 0.634i·23-s + (0.690 − 0.723i)25-s − 0.165·27-s − 1.30i·29-s − 1.79·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.206 + 0.978i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ -0.206 + 0.978i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5547520611\)
\(L(\frac12)\) \(\approx\) \(0.5547520611\)
\(L(\frac{7}{2})\) 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 \( 1 + (51.3 - 21.9i)T \)
good3 \( 1 - 1.29T + 243T^{2} \)
7 \( 1 - 170. iT - 1.68e4T^{2} \)
11 \( 1 + 39.8iT - 1.61e5T^{2} \)
13 \( 1 - 537.T + 3.71e5T^{2} \)
17 \( 1 + 1.35e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.03e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.61e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.91e3iT - 2.05e7T^{2} \)
31 \( 1 + 9.61e3T + 2.86e7T^{2} \)
37 \( 1 - 5.29e3T + 6.93e7T^{2} \)
41 \( 1 + 1.21e4T + 1.15e8T^{2} \)
43 \( 1 - 1.93e4T + 1.47e8T^{2} \)
47 \( 1 + 7.89e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.89e4T + 4.18e8T^{2} \)
59 \( 1 + 3.23e4iT - 7.14e8T^{2} \)
61 \( 1 + 4.88e4iT - 8.44e8T^{2} \)
67 \( 1 + 882.T + 1.35e9T^{2} \)
71 \( 1 - 8.81e3T + 1.80e9T^{2} \)
73 \( 1 - 1.43e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.98e4T + 3.07e9T^{2} \)
83 \( 1 - 1.39e4T + 3.93e9T^{2} \)
89 \( 1 - 7.58e3T + 5.58e9T^{2} \)
97 \( 1 + 6.67e4iT - 8.58e9T^{2} \)
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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

−11.42531940477535672288093196694, −11.22816784940065235089414996459, −9.397242512051188225574347222100, −8.628045556984779104749661383680, −7.63336306639372031254172157218, −6.25353485470660157643998362108, −5.19163331063169079349255209679, −3.52369935167508444771463751103, −2.46152659715180627362368129746, −0.19746806046688371849494377470, 1.23117550822730718099656160299, 3.43186248004962777500955010679, 4.24874877597939084392744975398, 5.79857505756040760009922231347, 7.17505511826713805127677828230, 8.146829946042718226409127241063, 8.961137424703899217972869441201, 10.60499807828295554571204204158, 11.05962123807369577457574529789, 12.31872861798688871182328716873

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