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Properties

Label 2-320-16.3-c2-0-1
Degree $2$
Conductor $320$
Sign $-0.910 - 0.414i$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.374 − 0.374i)3-s + (−1.58 + 1.58i)5-s − 2.42·7-s + 8.71i·9-s + (−13.8 − 13.8i)11-s + (8.90 + 8.90i)13-s + 1.18i·15-s − 27.7·17-s + (−7.49 + 7.49i)19-s + (−0.908 + 0.908i)21-s − 13.5·23-s − 5.00i·25-s + (6.64 + 6.64i)27-s + (−10.5 − 10.5i)29-s + 46.4i·31-s + ⋯
L(s)  = 1  + (0.124 − 0.124i)3-s + (−0.316 + 0.316i)5-s − 0.346·7-s + 0.968i·9-s + (−1.25 − 1.25i)11-s + (0.684 + 0.684i)13-s + 0.0790i·15-s − 1.63·17-s + (−0.394 + 0.394i)19-s + (−0.0432 + 0.0432i)21-s − 0.587·23-s − 0.200i·25-s + (0.246 + 0.246i)27-s + (−0.363 − 0.363i)29-s + 1.49i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.910 - 0.414i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.910 - 0.414i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0837686 + 0.386130i\)
\(L(\frac12)\) \(\approx\) \(0.0837686 + 0.386130i\)
\(L(2)\) 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 \( 1 + (1.58 - 1.58i)T \)
good3 \( 1 + (-0.374 + 0.374i)T - 9iT^{2} \)
7 \( 1 + 2.42T + 49T^{2} \)
11 \( 1 + (13.8 + 13.8i)T + 121iT^{2} \)
13 \( 1 + (-8.90 - 8.90i)T + 169iT^{2} \)
17 \( 1 + 27.7T + 289T^{2} \)
19 \( 1 + (7.49 - 7.49i)T - 361iT^{2} \)
23 \( 1 + 13.5T + 529T^{2} \)
29 \( 1 + (10.5 + 10.5i)T + 841iT^{2} \)
31 \( 1 - 46.4iT - 961T^{2} \)
37 \( 1 + (4.68 - 4.68i)T - 1.36e3iT^{2} \)
41 \( 1 - 38.6iT - 1.68e3T^{2} \)
43 \( 1 + (45.6 + 45.6i)T + 1.84e3iT^{2} \)
47 \( 1 - 35.3iT - 2.20e3T^{2} \)
53 \( 1 + (-61.0 + 61.0i)T - 2.80e3iT^{2} \)
59 \( 1 + (-9.21 - 9.21i)T + 3.48e3iT^{2} \)
61 \( 1 + (-20.3 - 20.3i)T + 3.72e3iT^{2} \)
67 \( 1 + (3.47 - 3.47i)T - 4.48e3iT^{2} \)
71 \( 1 - 8.72T + 5.04e3T^{2} \)
73 \( 1 + 23.1iT - 5.32e3T^{2} \)
79 \( 1 - 73.9iT - 6.24e3T^{2} \)
83 \( 1 + (-50.7 + 50.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 51.4iT - 7.92e3T^{2} \)
97 \( 1 - 82.8T + 9.40e3T^{2} \)
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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

−11.55807421694307506726685993376, −10.91344133085930911479895920741, −10.20733496603390807455225415194, −8.675502087034370779256325526718, −8.222354165935539787642067921622, −6.99469260975829561300832423310, −6.01483401267376590629386004480, −4.77564641010698956225989873484, −3.42514150413620778135468301106, −2.16191566048571487599682153650, 0.17030989132713534109475283665, 2.32515317045796398990373904229, 3.78346823578789146878948060940, 4.81940356249997752887474200612, 6.12720581283959257045516832552, 7.16481668208220797469323705558, 8.226974909759208787300193386667, 9.149590647623436009165120661122, 10.04579095813242975543367973852, 10.96368395919074072096138350881

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