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Properties

Label 2-160-20.7-c5-0-14
Degree $2$
Conductor $160$
Sign $-0.0151 + 0.999i$
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  + (−13.0 − 13.0i)3-s + (−30.1 + 47.1i)5-s + (−150. + 150. i)7-s + 97.6i·9-s + 579. i·11-s + (649. − 649. i)13-s + (1.00e3 − 221. i)15-s + (−106. − 106. i)17-s − 1.20e3·19-s + 3.93e3·21-s + (−609. − 609. i)23-s + (−1.31e3 − 2.83e3i)25-s + (−1.89e3 + 1.89e3i)27-s + 6.01e3i·29-s − 6.41e3i·31-s + ⋯
L(s)  = 1  + (−0.837 − 0.837i)3-s + (−0.538 + 0.842i)5-s + (−1.16 + 1.16i)7-s + 0.401i·9-s + 1.44i·11-s + (1.06 − 1.06i)13-s + (1.15 − 0.254i)15-s + (−0.0893 − 0.0893i)17-s − 0.766·19-s + 1.94·21-s + (−0.240 − 0.240i)23-s + (−0.419 − 0.907i)25-s + (−0.500 + 0.500i)27-s + 1.32i·29-s − 1.19i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0151 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0151 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.0151 + 0.999i$
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: \((2,\ 160,\ (\ :5/2),\ -0.0151 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4659482520\)
\(L(\frac12)\) \(\approx\) \(0.4659482520\)
\(L(\frac{7}{2})\) 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 \( 1 + (30.1 - 47.1i)T \)
good3 \( 1 + (13.0 + 13.0i)T + 243iT^{2} \)
7 \( 1 + (150. - 150. i)T - 1.68e4iT^{2} \)
11 \( 1 - 579. iT - 1.61e5T^{2} \)
13 \( 1 + (-649. + 649. i)T - 3.71e5iT^{2} \)
17 \( 1 + (106. + 106. i)T + 1.41e6iT^{2} \)
19 \( 1 + 1.20e3T + 2.47e6T^{2} \)
23 \( 1 + (609. + 609. i)T + 6.43e6iT^{2} \)
29 \( 1 - 6.01e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.41e3iT - 2.86e7T^{2} \)
37 \( 1 + (2.85e3 + 2.85e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.29e4T + 1.15e8T^{2} \)
43 \( 1 + (-3.19e3 - 3.19e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-1.49e4 + 1.49e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-2.35e4 + 2.35e4i)T - 4.18e8iT^{2} \)
59 \( 1 + 1.69e4T + 7.14e8T^{2} \)
61 \( 1 - 5.12e4T + 8.44e8T^{2} \)
67 \( 1 + (4.75e3 - 4.75e3i)T - 1.35e9iT^{2} \)
71 \( 1 + 9.19e3iT - 1.80e9T^{2} \)
73 \( 1 + (-2.94e4 + 2.94e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 2.10e4T + 3.07e9T^{2} \)
83 \( 1 + (-2.18e4 - 2.18e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 1.02e4iT - 5.58e9T^{2} \)
97 \( 1 + (5.40e4 + 5.40e4i)T + 8.58e9iT^{2} \)
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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

−11.98690936172500359298925315426, −10.87185494045617618044279637870, −9.869559617281056144379768528314, −8.516276755518286087779305793295, −7.14096463815347742669349130353, −6.46973740156459771824805691808, −5.55152893747986870988330725301, −3.60218837532839869001986466152, −2.22061783554285876300289498037, −0.23846710952449167266891814006, 0.857541913037556255827029404673, 3.67792592261204887752704619316, 4.26591722771907366139394828554, 5.73099200602469481674472775612, 6.68910465258896302984579151833, 8.275568384922189380757967955882, 9.268082297294587024368040025951, 10.43151076505880111696754031801, 11.09808273872815464751231602424, 12.02569492009913676630079481153

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