Properties

Label 2-160-5.4-c5-0-28
Degree $2$
Conductor $160$
Sign $-0.485 - 0.874i$
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  − 26.2i·3-s + (48.8 − 27.1i)5-s + 23.9i·7-s − 447.·9-s − 704.·11-s + 1.03e3i·13-s + (−713. − 1.28e3i)15-s − 217. i·17-s − 1.29e3·19-s + 628.·21-s − 2.80e3i·23-s + (1.65e3 − 2.65e3i)25-s + 5.36e3i·27-s − 4.90e3·29-s − 415.·31-s + ⋯
L(s)  = 1  − 1.68i·3-s + (0.874 − 0.485i)5-s + 0.184i·7-s − 1.84·9-s − 1.75·11-s + 1.70i·13-s + (−0.818 − 1.47i)15-s − 0.182i·17-s − 0.819·19-s + 0.311·21-s − 1.10i·23-s + (0.528 − 0.849i)25-s + 1.41i·27-s − 1.08·29-s − 0.0776·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.485 - 0.874i$
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: \((2,\ 160,\ (\ :5/2),\ -0.485 - 0.874i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4524778981\)
\(L(\frac12)\) \(\approx\) \(0.4524778981\)
\(L(\frac{7}{2})\) 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 \( 1 + (-48.8 + 27.1i)T \)
good3 \( 1 + 26.2iT - 243T^{2} \)
7 \( 1 - 23.9iT - 1.68e4T^{2} \)
11 \( 1 + 704.T + 1.61e5T^{2} \)
13 \( 1 - 1.03e3iT - 3.71e5T^{2} \)
17 \( 1 + 217. iT - 1.41e6T^{2} \)
19 \( 1 + 1.29e3T + 2.47e6T^{2} \)
23 \( 1 + 2.80e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.90e3T + 2.05e7T^{2} \)
31 \( 1 + 415.T + 2.86e7T^{2} \)
37 \( 1 - 6.45e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.64e4T + 1.15e8T^{2} \)
43 \( 1 - 1.37e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.29e4iT - 2.29e8T^{2} \)
53 \( 1 + 5.23e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.40e4T + 7.14e8T^{2} \)
61 \( 1 + 1.03e3T + 8.44e8T^{2} \)
67 \( 1 + 4.79e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.47e4T + 1.80e9T^{2} \)
73 \( 1 + 2.27e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.42e4T + 3.07e9T^{2} \)
83 \( 1 + 2.25e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.78e4T + 5.58e9T^{2} \)
97 \( 1 + 8.41e4iT - 8.58e9T^{2} \)
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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

−11.62200259239647762724669095347, −10.34774117735005979061035383439, −8.963349509549293381657959485938, −8.126837888802368396124647727760, −6.94944339939491738916020202562, −6.11571835263399871312512753877, −4.92048435777121072920779766738, −2.46151803208059319307785816716, −1.74583305100143525204530096768, −0.13357109662032687206075867166, 2.57080955443150374792066600907, 3.62360032487593960610232107051, 5.25261260286411551591113747793, 5.65018164058507792122257378531, 7.59417508110628343891907202158, 8.812994984991626008192708359761, 10.04792757167686880544323252281, 10.39244493579380310380449687638, 11.06048085948890916842563428781, 12.87132038389746153759070213321

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