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Properties

Label 2-160-5.4-c5-0-14
Degree $2$
Conductor $160$
Sign $0.679 - 0.733i$
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  + (41 + 38i)5-s + 243·9-s − 244i·13-s + 808i·17-s + (237 + 3.11e3i)25-s + 2.95e3·29-s + 1.12e4i·37-s + 2.09e4·41-s + (9.96e3 + 9.23e3i)45-s + 1.68e4·49-s + 4.02e4i·53-s − 1.89e4·61-s + (9.27e3 − 1.00e4i)65-s + 2.01e4i·73-s + 5.90e4·81-s + ⋯
L(s)  = 1  + (0.733 + 0.679i)5-s + 9-s − 0.400i·13-s + 0.678i·17-s + (0.0758 + 0.997i)25-s + 0.651·29-s + 1.35i·37-s + 1.94·41-s + (0.733 + 0.679i)45-s + 49-s + 1.96i·53-s − 0.652·61-s + (0.272 − 0.293i)65-s + 0.442i·73-s + 0.999·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.679 - 0.733i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.679 - 0.733i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.375274729\)
\(L(\frac12)\) \(\approx\) \(2.375274729\)
\(L(\frac{7}{2})\) 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 \( 1 + (-41 - 38i)T \)
good3 \( 1 - 243T^{2} \)
7 \( 1 - 1.68e4T^{2} \)
11 \( 1 + 1.61e5T^{2} \)
13 \( 1 + 244iT - 3.71e5T^{2} \)
17 \( 1 - 808iT - 1.41e6T^{2} \)
19 \( 1 + 2.47e6T^{2} \)
23 \( 1 - 6.43e6T^{2} \)
29 \( 1 - 2.95e3T + 2.05e7T^{2} \)
31 \( 1 + 2.86e7T^{2} \)
37 \( 1 - 1.12e4iT - 6.93e7T^{2} \)
41 \( 1 - 2.09e4T + 1.15e8T^{2} \)
43 \( 1 - 1.47e8T^{2} \)
47 \( 1 - 2.29e8T^{2} \)
53 \( 1 - 4.02e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 + 1.89e4T + 8.44e8T^{2} \)
67 \( 1 - 1.35e9T^{2} \)
71 \( 1 + 1.80e9T^{2} \)
73 \( 1 - 2.01e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.07e9T^{2} \)
83 \( 1 - 3.93e9T^{2} \)
89 \( 1 - 5.10e4T + 5.58e9T^{2} \)
97 \( 1 + 1.60e5iT - 8.58e9T^{2} \)
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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

−12.24053416279030974616005693235, −10.83928606162156262006041393656, −10.20435932423617503199160699424, −9.239136980677900768343862971462, −7.82518586435054976541628980654, −6.76488068007913051170517762209, −5.76230222537753459485290924106, −4.28360794747669414459934651820, −2.77960580556261750667077531932, −1.34259265072254871265100399845, 0.898288736146675267206550494747, 2.24149882851433345226224630586, 4.13938507608871584646647187054, 5.21940610128131324596662538955, 6.48229949522269817301129805731, 7.63259014263448549602639753118, 8.991627646104298067621072039041, 9.699601943521478162833043535403, 10.72356395111951850977351730825, 12.04805710887510773441566887206

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