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Properties

Label 2-320-40.19-c2-0-4
Degree $2$
Conductor $320$
Sign $-0.945 + 0.325i$
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  + 5.13i·3-s + (2.79 + 4.14i)5-s − 6.19·7-s − 17.3·9-s − 20.0·11-s + 15.8·13-s + (−21.2 + 14.3i)15-s − 6.98i·17-s + 10.3·19-s − 31.7i·21-s + 22.3·23-s + (−9.35 + 23.1i)25-s − 42.9i·27-s + 4.20i·29-s + 20.7i·31-s + ⋯
L(s)  = 1  + 1.71i·3-s + (0.559 + 0.828i)5-s − 0.884·7-s − 1.92·9-s − 1.82·11-s + 1.22·13-s + (−1.41 + 0.957i)15-s − 0.411i·17-s + 0.546·19-s − 1.51i·21-s + 0.973·23-s + (−0.374 + 0.927i)25-s − 1.58i·27-s + 0.145i·29-s + 0.668i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.945 + 0.325i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.945 + 0.325i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.172381 - 1.02960i\)
\(L(\frac12)\) \(\approx\) \(0.172381 - 1.02960i\)
\(L(2)\) 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 \( 1 + (-2.79 - 4.14i)T \)
good3 \( 1 - 5.13iT - 9T^{2} \)
7 \( 1 + 6.19T + 49T^{2} \)
11 \( 1 + 20.0T + 121T^{2} \)
13 \( 1 - 15.8T + 169T^{2} \)
17 \( 1 + 6.98iT - 289T^{2} \)
19 \( 1 - 10.3T + 361T^{2} \)
23 \( 1 - 22.3T + 529T^{2} \)
29 \( 1 - 4.20iT - 841T^{2} \)
31 \( 1 - 20.7iT - 961T^{2} \)
37 \( 1 + 35.4T + 1.36e3T^{2} \)
41 \( 1 + 37.0T + 1.68e3T^{2} \)
43 \( 1 - 23.8iT - 1.84e3T^{2} \)
47 \( 1 + 48.7T + 2.20e3T^{2} \)
53 \( 1 - 77.4T + 2.80e3T^{2} \)
59 \( 1 + 0.497T + 3.48e3T^{2} \)
61 \( 1 - 60.7iT - 3.72e3T^{2} \)
67 \( 1 - 82.5iT - 4.48e3T^{2} \)
71 \( 1 + 28.7iT - 5.04e3T^{2} \)
73 \( 1 - 10.1iT - 5.32e3T^{2} \)
79 \( 1 + 87.5iT - 6.24e3T^{2} \)
83 \( 1 - 103. iT - 6.88e3T^{2} \)
89 \( 1 + 49.2T + 7.92e3T^{2} \)
97 \( 1 - 84.4iT - 9.40e3T^{2} \)
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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

−11.47316046039580205322564298598, −10.59498518850492140678251992257, −10.26073100253873371445455809374, −9.415423415723916316475320443178, −8.467551053811917067577695489203, −6.98837303493868824848436227478, −5.74894690542684368189273744945, −5.02650920361536973697797579898, −3.45534468692820931830472394650, −2.88386940538432660356771794683, 0.47408398635801451360214302306, 1.84747417330236240268743752510, 3.12560070458181100378985430220, 5.26689504799084556521035859965, 6.03819345792459771285919071499, 6.98565661262022128107599822926, 8.062973602393676106550322623811, 8.685880342945264333527908584315, 9.936484045426286889241774736363, 11.03703927980182625345964218169

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