Properties

Label 2-320-40.19-c2-0-0
Degree $2$
Conductor $320$
Sign $-0.721 - 0.692i$
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  − 1.90i·3-s + (−4.37 − 2.41i)5-s − 3.95·7-s + 5.35·9-s − 6.18·11-s − 4.94·13-s + (−4.60 + 8.35i)15-s + 22.4i·17-s − 10.3·19-s + 7.54i·21-s − 39.6·23-s + (13.3 + 21.1i)25-s − 27.4i·27-s + 30.4i·29-s + 24.7i·31-s + ⋯
L(s)  = 1  − 0.636i·3-s + (−0.875 − 0.482i)5-s − 0.565·7-s + 0.595·9-s − 0.562·11-s − 0.380·13-s + (−0.306 + 0.557i)15-s + 1.31i·17-s − 0.546·19-s + 0.359i·21-s − 1.72·23-s + (0.534 + 0.845i)25-s − 1.01i·27-s + 1.04i·29-s + 0.797i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.721 - 0.692i$
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: \((2,\ 320,\ (\ :1),\ -0.721 - 0.692i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0149546 + 0.0371528i\)
\(L(\frac12)\) \(\approx\) \(0.0149546 + 0.0371528i\)
\(L(2)\) 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 \( 1 + (4.37 + 2.41i)T \)
good3 \( 1 + 1.90iT - 9T^{2} \)
7 \( 1 + 3.95T + 49T^{2} \)
11 \( 1 + 6.18T + 121T^{2} \)
13 \( 1 + 4.94T + 169T^{2} \)
17 \( 1 - 22.4iT - 289T^{2} \)
19 \( 1 + 10.3T + 361T^{2} \)
23 \( 1 + 39.6T + 529T^{2} \)
29 \( 1 - 30.4iT - 841T^{2} \)
31 \( 1 - 24.7iT - 961T^{2} \)
37 \( 1 + 24.0T + 1.36e3T^{2} \)
41 \( 1 - 31.0T + 1.68e3T^{2} \)
43 \( 1 + 52.2iT - 1.84e3T^{2} \)
47 \( 1 + 13.1T + 2.20e3T^{2} \)
53 \( 1 - 17.9T + 2.80e3T^{2} \)
59 \( 1 + 104.T + 3.48e3T^{2} \)
61 \( 1 + 57.2iT - 3.72e3T^{2} \)
67 \( 1 - 99.3iT - 4.48e3T^{2} \)
71 \( 1 + 16.7iT - 5.04e3T^{2} \)
73 \( 1 + 96.3iT - 5.32e3T^{2} \)
79 \( 1 + 139. iT - 6.24e3T^{2} \)
83 \( 1 - 91.7iT - 6.88e3T^{2} \)
89 \( 1 + 94.7T + 7.92e3T^{2} \)
97 \( 1 + 143. iT - 9.40e3T^{2} \)
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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

−12.20731733434307089888929748089, −10.79006586497478318560855163801, −10.02313652089219074028009479859, −8.718396210646385904574471998915, −7.919863571511493803347209195199, −7.08063589885985584485114858300, −6.02072911036254329483971057952, −4.60652418935913305015261308651, −3.53888745767984038227408704895, −1.77404933418191759674797433764, 0.01829118253767435364588827431, 2.62833269702359924076322423650, 3.87848038995359616507473042669, 4.72693484541851779709617991042, 6.19915084861525754263900868824, 7.31752540918733266562298553883, 8.058232773722490546965078854542, 9.495741905168625797444037465976, 10.05587395821094842814819089410, 11.00398853278766490273457225152

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