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Properties

Label 2-320-16.3-c2-0-2
Degree $2$
Conductor $320$
Sign $-0.902 - 0.429i$
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  + (−2.58 + 2.58i)3-s + (1.58 − 1.58i)5-s − 0.523·7-s − 4.32i·9-s + (4.46 + 4.46i)11-s + (11.7 + 11.7i)13-s + 8.16i·15-s − 23.7·17-s + (−13.0 + 13.0i)19-s + (1.35 − 1.35i)21-s − 31.0·23-s − 5.00i·25-s + (−12.0 − 12.0i)27-s + (11.4 + 11.4i)29-s − 29.5i·31-s + ⋯
L(s)  = 1  + (−0.860 + 0.860i)3-s + (0.316 − 0.316i)5-s − 0.0748·7-s − 0.480i·9-s + (0.406 + 0.406i)11-s + (0.905 + 0.905i)13-s + 0.544i·15-s − 1.39·17-s + (−0.684 + 0.684i)19-s + (0.0643 − 0.0643i)21-s − 1.35·23-s − 0.200i·25-s + (−0.447 − 0.447i)27-s + (0.395 + 0.395i)29-s − 0.952i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.429i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.902 - 0.429i$
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: \((2,\ 320,\ (\ :1),\ -0.902 - 0.429i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.161266 + 0.714037i\)
\(L(\frac12)\) \(\approx\) \(0.161266 + 0.714037i\)
\(L(2)\) 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 \( 1 + (2.58 - 2.58i)T - 9iT^{2} \)
7 \( 1 + 0.523T + 49T^{2} \)
11 \( 1 + (-4.46 - 4.46i)T + 121iT^{2} \)
13 \( 1 + (-11.7 - 11.7i)T + 169iT^{2} \)
17 \( 1 + 23.7T + 289T^{2} \)
19 \( 1 + (13.0 - 13.0i)T - 361iT^{2} \)
23 \( 1 + 31.0T + 529T^{2} \)
29 \( 1 + (-11.4 - 11.4i)T + 841iT^{2} \)
31 \( 1 + 29.5iT - 961T^{2} \)
37 \( 1 + (43.0 - 43.0i)T - 1.36e3iT^{2} \)
41 \( 1 + 25.9iT - 1.68e3T^{2} \)
43 \( 1 + (-11.0 - 11.0i)T + 1.84e3iT^{2} \)
47 \( 1 - 72.9iT - 2.20e3T^{2} \)
53 \( 1 + (49.4 - 49.4i)T - 2.80e3iT^{2} \)
59 \( 1 + (-31.1 - 31.1i)T + 3.48e3iT^{2} \)
61 \( 1 + (-13.9 - 13.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (-66.6 + 66.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 34.5T + 5.04e3T^{2} \)
73 \( 1 - 101. iT - 5.32e3T^{2} \)
79 \( 1 - 60.9iT - 6.24e3T^{2} \)
83 \( 1 + (-33.6 + 33.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 95.8iT - 7.92e3T^{2} \)
97 \( 1 - 135.T + 9.40e3T^{2} \)
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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

−11.60414061782625635729187212319, −10.91046163216272165774477044587, −10.02771426020353842155584679870, −9.204921274667346811139986468272, −8.223016540657632095271515714189, −6.59173489782870531908173103041, −5.95547563649077069105179304192, −4.63603192044263756901072990291, −4.01570532865149671989435460714, −1.87236653435798028659012358748, 0.37497378459309371295256431288, 1.98468559874004870123484178754, 3.67609114713647212668065443351, 5.27527928135255368064838877701, 6.37551789549433576422425093589, 6.69687739930249830273362827320, 8.106022821351280071817812893578, 9.038049080199096965441180638223, 10.40233857734576792936421269273, 11.09802316907695567807032858694

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