Properties

Label 2-320-5.4-c5-0-8
Degree $2$
Conductor $320$
Sign $-0.221 + 0.975i$
Analytic cond. $51.3228$
Root an. cond. $7.16399$
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  + 16.9i·3-s + (−12.3 + 54.5i)5-s + 211. i·7-s − 45.4·9-s − 520.·11-s − 732. i·13-s + (−925. − 210. i)15-s + 2.26e3i·17-s − 2.03e3·19-s − 3.59e3·21-s − 974. i·23-s + (−2.81e3 − 1.35e3i)25-s + 3.35e3i·27-s + 5.27e3·29-s + 2.00e3·31-s + ⋯
L(s)  = 1  + 1.08i·3-s + (−0.221 + 0.975i)5-s + 1.63i·7-s − 0.186·9-s − 1.29·11-s − 1.20i·13-s + (−1.06 − 0.241i)15-s + 1.90i·17-s − 1.29·19-s − 1.77·21-s − 0.384i·23-s + (−0.901 − 0.432i)25-s + 0.885i·27-s + 1.16·29-s + 0.374·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ -0.221 + 0.975i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9511202659\)
\(L(\frac12)\) \(\approx\) \(0.9511202659\)
\(L(\frac{7}{2})\) 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 \( 1 + (12.3 - 54.5i)T \)
good3 \( 1 - 16.9iT - 243T^{2} \)
7 \( 1 - 211. iT - 1.68e4T^{2} \)
11 \( 1 + 520.T + 1.61e5T^{2} \)
13 \( 1 + 732. iT - 3.71e5T^{2} \)
17 \( 1 - 2.26e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.03e3T + 2.47e6T^{2} \)
23 \( 1 + 974. iT - 6.43e6T^{2} \)
29 \( 1 - 5.27e3T + 2.05e7T^{2} \)
31 \( 1 - 2.00e3T + 2.86e7T^{2} \)
37 \( 1 - 3.65e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.71e4T + 1.15e8T^{2} \)
43 \( 1 - 4.07e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.33e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.74e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.89e3T + 7.14e8T^{2} \)
61 \( 1 + 8.73e3T + 8.44e8T^{2} \)
67 \( 1 - 4.09e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.68e4T + 1.80e9T^{2} \)
73 \( 1 + 9.82e3iT - 2.07e9T^{2} \)
79 \( 1 - 6.03e4T + 3.07e9T^{2} \)
83 \( 1 + 2.41e3iT - 3.93e9T^{2} \)
89 \( 1 + 6.93e4T + 5.58e9T^{2} \)
97 \( 1 + 7.62e4iT - 8.58e9T^{2} \)
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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

−11.05439212969052698194308020976, −10.51577552502882959122763932622, −9.902446485915983567669342475572, −8.533323888443356762728647750248, −8.017809361871125398074701201173, −6.32584951141547630228438464067, −5.58987183067402318391613669617, −4.42341173762134746929482278571, −3.13540065974709898775412399671, −2.32520833178898893449799360115, 0.28139434845472859232736966640, 1.03251431655905680565838913091, 2.34844046212926818003852036256, 4.16343149945734517591673071165, 4.90805109768856136321911537507, 6.50661609001907851788120241212, 7.37989083323089430877274140439, 7.87622180558621879953545509586, 9.095915270628078094285124757756, 10.17098621156149274036331725972

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