Properties

Label 2-160-8.5-c5-0-7
Degree $2$
Conductor $160$
Sign $0.999 + 0.00874i$
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  − 10.7i·3-s + 25i·5-s − 198.·7-s + 127.·9-s − 85.9i·11-s + 407. i·13-s + 268.·15-s + 1.20e3·17-s + 206. i·19-s + 2.13e3i·21-s + 2.59e3·23-s − 625·25-s − 3.98e3i·27-s + 6.19e3i·29-s + 1.86e3·31-s + ⋯
L(s)  = 1  − 0.689i·3-s + 0.447i·5-s − 1.53·7-s + 0.524·9-s − 0.214i·11-s + 0.668i·13-s + 0.308·15-s + 1.01·17-s + 0.130i·19-s + 1.05i·21-s + 1.02·23-s − 0.200·25-s − 1.05i·27-s + 1.36i·29-s + 0.348·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00874i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.999 + 0.00874i$
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: \((2,\ 160,\ (\ :5/2),\ 0.999 + 0.00874i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.667163062\)
\(L(\frac12)\) \(\approx\) \(1.667163062\)
\(L(\frac{7}{2})\) 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 \( 1 + 10.7iT - 243T^{2} \)
7 \( 1 + 198.T + 1.68e4T^{2} \)
11 \( 1 + 85.9iT - 1.61e5T^{2} \)
13 \( 1 - 407. iT - 3.71e5T^{2} \)
17 \( 1 - 1.20e3T + 1.41e6T^{2} \)
19 \( 1 - 206. iT - 2.47e6T^{2} \)
23 \( 1 - 2.59e3T + 6.43e6T^{2} \)
29 \( 1 - 6.19e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.86e3T + 2.86e7T^{2} \)
37 \( 1 + 1.47e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.80e4T + 1.15e8T^{2} \)
43 \( 1 - 9.26e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.43e4T + 2.29e8T^{2} \)
53 \( 1 + 1.27e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.07e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.13e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.26e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.12e4T + 1.80e9T^{2} \)
73 \( 1 - 2.32e4T + 2.07e9T^{2} \)
79 \( 1 + 2.91e4T + 3.07e9T^{2} \)
83 \( 1 + 4.80e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.01e4T + 5.58e9T^{2} \)
97 \( 1 + 1.13e5T + 8.58e9T^{2} \)
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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

−12.33859847989328198086651597340, −10.94802593519368552729879820308, −9.912202452287071334504511577129, −9.040371988768907034432907372800, −7.43421330043882178702813393979, −6.81202817738313948033692251719, −5.78174481872371555955247518178, −3.88348320006659982165394315419, −2.66962030976711119621374947476, −0.949451348844032444099508165619, 0.76310990282658015833754615043, 2.95057851242125985210581669047, 4.07282944184490804779821144625, 5.37095869104114946240195014938, 6.58441175179788059363841361262, 7.83213943308547388004150314474, 9.311096869359012516996615088728, 9.811180884866177018681842435054, 10.70295865660830849785010673604, 12.23147225659268308547253070728

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