Properties

Label 2-320-16.13-c5-0-0
Degree $2$
Conductor $320$
Sign $-0.827 + 0.561i$
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  + (−21.2 − 21.2i)3-s + (−17.6 + 17.6i)5-s + 169. i·7-s + 660. i·9-s + (−379. + 379. i)11-s + (10.8 + 10.8i)13-s + 751.·15-s − 1.78e3·17-s + (1.09e3 + 1.09e3i)19-s + (3.60e3 − 3.60e3i)21-s + 2.09e3i·23-s − 625i·25-s + (8.86e3 − 8.86e3i)27-s + (100. + 100. i)29-s − 3.15e3·31-s + ⋯
L(s)  = 1  + (−1.36 − 1.36i)3-s + (−0.316 + 0.316i)5-s + 1.30i·7-s + 2.71i·9-s + (−0.944 + 0.944i)11-s + (0.0177 + 0.0177i)13-s + 0.862·15-s − 1.50·17-s + (0.694 + 0.694i)19-s + (1.78 − 1.78i)21-s + 0.826i·23-s − 0.200i·25-s + (2.34 − 2.34i)27-s + (0.0221 + 0.0221i)29-s − 0.589·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.827 + 0.561i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ -0.827 + 0.561i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.09644201387\)
\(L(\frac12)\) \(\approx\) \(0.09644201387\)
\(L(\frac{7}{2})\) 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 \( 1 + (17.6 - 17.6i)T \)
good3 \( 1 + (21.2 + 21.2i)T + 243iT^{2} \)
7 \( 1 - 169. iT - 1.68e4T^{2} \)
11 \( 1 + (379. - 379. i)T - 1.61e5iT^{2} \)
13 \( 1 + (-10.8 - 10.8i)T + 3.71e5iT^{2} \)
17 \( 1 + 1.78e3T + 1.41e6T^{2} \)
19 \( 1 + (-1.09e3 - 1.09e3i)T + 2.47e6iT^{2} \)
23 \( 1 - 2.09e3iT - 6.43e6T^{2} \)
29 \( 1 + (-100. - 100. i)T + 2.05e7iT^{2} \)
31 \( 1 + 3.15e3T + 2.86e7T^{2} \)
37 \( 1 + (-7.28e3 + 7.28e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.30e4iT - 1.15e8T^{2} \)
43 \( 1 + (1.35e3 - 1.35e3i)T - 1.47e8iT^{2} \)
47 \( 1 + 1.84e4T + 2.29e8T^{2} \)
53 \( 1 + (1.01e4 - 1.01e4i)T - 4.18e8iT^{2} \)
59 \( 1 + (5.69e3 - 5.69e3i)T - 7.14e8iT^{2} \)
61 \( 1 + (1.68e4 + 1.68e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (8.86e3 + 8.86e3i)T + 1.35e9iT^{2} \)
71 \( 1 - 2.31e4iT - 1.80e9T^{2} \)
73 \( 1 - 134. iT - 2.07e9T^{2} \)
79 \( 1 - 2.69e4T + 3.07e9T^{2} \)
83 \( 1 + (-2.02e4 - 2.02e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.01e5iT - 5.58e9T^{2} \)
97 \( 1 + 4.78e4T + 8.58e9T^{2} \)
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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

−11.45266688283694889002763488074, −10.87787699164344508613290381190, −9.573315207953826467118095716859, −8.124192783278745944957359888618, −7.42149786111252832041443840485, −6.46175795871044656148838862777, −5.62618798403695028473574263560, −4.78848623922744374938810259620, −2.55241555718147493808055228013, −1.67374881258008566196313711949, 0.04982634003135695414663447961, 0.66358457011250492476929374278, 3.31612070284328308906881372917, 4.38074985189316419111922875682, 4.96745436340452371317189167275, 6.12009061671007823225453553144, 7.11652253615012551290112268278, 8.548335039187956351991667832089, 9.582117254199453978663018920192, 10.57242120999478914054027081246

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