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Properties

Label 2-320-20.19-c2-0-21
Degree $2$
Conductor $320$
Sign $-0.200 + 0.979i$
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  + 2.82·3-s + (1 − 4.89i)5-s − 8.48·7-s − 0.999·9-s − 13.8i·11-s − 9.79i·13-s + (2.82 − 13.8i)15-s − 19.5i·17-s + 13.8i·19-s − 24·21-s + 25.4·23-s + (−22.9 − 9.79i)25-s − 28.2·27-s + 22·29-s + 55.4i·31-s + ⋯
L(s)  = 1  + 0.942·3-s + (0.200 − 0.979i)5-s − 1.21·7-s − 0.111·9-s − 1.25i·11-s − 0.753i·13-s + (0.188 − 0.923i)15-s − 1.15i·17-s + 0.729i·19-s − 1.14·21-s + 1.10·23-s + (−0.919 − 0.391i)25-s − 1.04·27-s + 0.758·29-s + 1.78i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.200 + 0.979i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.200 + 0.979i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05963 - 1.29777i\)
\(L(\frac12)\) \(\approx\) \(1.05963 - 1.29777i\)
\(L(2)\) 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 \( 1 + (-1 + 4.89i)T \)
good3 \( 1 - 2.82T + 9T^{2} \)
7 \( 1 + 8.48T + 49T^{2} \)
11 \( 1 + 13.8iT - 121T^{2} \)
13 \( 1 + 9.79iT - 169T^{2} \)
17 \( 1 + 19.5iT - 289T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
23 \( 1 - 25.4T + 529T^{2} \)
29 \( 1 - 22T + 841T^{2} \)
31 \( 1 - 55.4iT - 961T^{2} \)
37 \( 1 + 48.9iT - 1.36e3T^{2} \)
41 \( 1 - 22T + 1.68e3T^{2} \)
43 \( 1 - 59.3T + 1.84e3T^{2} \)
47 \( 1 + 8.48T + 2.20e3T^{2} \)
53 \( 1 + 29.3iT - 2.80e3T^{2} \)
59 \( 1 + 13.8iT - 3.48e3T^{2} \)
61 \( 1 + 46T + 3.72e3T^{2} \)
67 \( 1 - 59.3T + 4.48e3T^{2} \)
71 \( 1 + 27.7iT - 5.04e3T^{2} \)
73 \( 1 - 78.3iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 76.3T + 6.88e3T^{2} \)
89 \( 1 - 146T + 7.92e3T^{2} \)
97 \( 1 + 58.7iT - 9.40e3T^{2} \)
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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

−11.11204181579719506087519773843, −9.951770091008049668263259539264, −9.047121071005775584679134591450, −8.617605881333376923983838689211, −7.51516812199444258865857953719, −6.13419844434145360874092767053, −5.20733233549774802407406216889, −3.54659381629903562490813600649, −2.78343640777394687265652041720, −0.69232352531786258596999828554, 2.23925253579375654766337338475, 3.11068996187990201797208210023, 4.27057742515754078824103503426, 6.09985406820573225134709585157, 6.86627521103266286922015717650, 7.78738775676437303369423557440, 9.122570353523429616917649970867, 9.637607274171719904817151551131, 10.56780300829810769505506459826, 11.64198427180637211848143308698

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