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Properties

Label 2-160-5.4-c5-0-9
Degree $2$
Conductor $160$
Sign $0.942 - 0.335i$
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  − 21.9i·3-s + (18.7 + 52.6i)5-s − 29.6i·7-s − 239.·9-s + 227.·11-s + 1.06e3i·13-s + (1.15e3 − 412. i)15-s + 686. i·17-s − 1.30e3·19-s − 652.·21-s + 4.07e3i·23-s + (−2.42e3 + 1.97e3i)25-s − 78.9i·27-s + 4.53e3·29-s + 7.69e3·31-s + ⋯
L(s)  = 1  − 1.40i·3-s + (0.335 + 0.942i)5-s − 0.229i·7-s − 0.985·9-s + 0.565·11-s + 1.75i·13-s + (1.32 − 0.472i)15-s + 0.575i·17-s − 0.831·19-s − 0.322·21-s + 1.60i·23-s + (−0.774 + 0.632i)25-s − 0.0208i·27-s + 1.00·29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.942 - 0.335i$
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: \((2,\ 160,\ (\ :5/2),\ 0.942 - 0.335i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.884515648\)
\(L(\frac12)\) \(\approx\) \(1.884515648\)
\(L(\frac{7}{2})\) 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 \( 1 + (-18.7 - 52.6i)T \)
good3 \( 1 + 21.9iT - 243T^{2} \)
7 \( 1 + 29.6iT - 1.68e4T^{2} \)
11 \( 1 - 227.T + 1.61e5T^{2} \)
13 \( 1 - 1.06e3iT - 3.71e5T^{2} \)
17 \( 1 - 686. iT - 1.41e6T^{2} \)
19 \( 1 + 1.30e3T + 2.47e6T^{2} \)
23 \( 1 - 4.07e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.53e3T + 2.05e7T^{2} \)
31 \( 1 - 7.69e3T + 2.86e7T^{2} \)
37 \( 1 + 9.32e3iT - 6.93e7T^{2} \)
41 \( 1 - 3.75e3T + 1.15e8T^{2} \)
43 \( 1 - 2.93e3iT - 1.47e8T^{2} \)
47 \( 1 - 3.69e3iT - 2.29e8T^{2} \)
53 \( 1 + 4.81e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.11e4T + 7.14e8T^{2} \)
61 \( 1 + 3.38e4T + 8.44e8T^{2} \)
67 \( 1 - 2.48e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.41e4T + 1.80e9T^{2} \)
73 \( 1 - 7.31e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.36e4T + 3.07e9T^{2} \)
83 \( 1 + 6.73e4iT - 3.93e9T^{2} \)
89 \( 1 - 6.02e4T + 5.58e9T^{2} \)
97 \( 1 + 1.54e4iT - 8.58e9T^{2} \)
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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

−11.97118342171486005547314942303, −11.33915857518888690680782249462, −10.05725583162528575655298283964, −8.837805000586175439521176626052, −7.52917285880654034886984765033, −6.73538715807220150820408460470, −6.10765669644260603052461328626, −4.04477733771279983776002526605, −2.34908523879968849398026466432, −1.35740480538044642471475506766, 0.67249609560643556372348706082, 2.82650240861355470261679046819, 4.33385614927263842174828680695, 5.06862041610665247963336746640, 6.24537983091129880661263908358, 8.231097656102056724179426310834, 8.915885364476583211667681032349, 10.03463138123834909314365783352, 10.53116065712016163376399750966, 11.95904554092426749851809812254

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