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Properties

Label 2-320-40.19-c2-0-11
Degree $2$
Conductor $320$
Sign $0.325 - 0.945i$
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

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

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.325 - 0.945i$
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: \((2,\ 320,\ (\ :1),\ 0.325 - 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.72214 + 1.22818i\)
\(L(\frac12)\) \(\approx\) \(1.72214 + 1.22818i\)
\(L(2)\) 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

−11.42591789676525763912804805276, −10.56813471181173646876365744907, −9.714514255793918402155669112281, −8.763770230219774876337393293632, −8.490128105692209705826987864504, −6.36638471829190141392050078766, −5.43216852454460570301301785792, −4.34960592677895400621516786488, −3.77248759103101685324018208263, −1.55984119541742929820027764327, 1.28224537066818358078075610757, 2.13135147281065924604166437088, 3.78563157597260477431933449086, 5.76279554942017914236613877923, 6.50083845001037103546892204186, 7.16450124729999087034344284498, 8.269876988605635953985570607978, 9.067284942424216918305572857022, 10.55629477388141713421488514015, 11.59668019104090821381611780242

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