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Properties

Label 2-320-20.3-c1-0-7
Degree $2$
Conductor $320$
Sign $0.525 + 0.850i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − 1.73i)3-s + (1 − 2i)5-s + (1.73 + 1.73i)7-s − 2.99i·9-s + 3.46i·11-s + (−1 − i)13-s + (−1.73 − 5.19i)15-s + (1 − i)17-s − 6.92·19-s + 5.99·21-s + (−1.73 + 1.73i)23-s + (−3 − 4i)25-s − 4i·29-s + 3.46i·31-s + (5.99 + 5.99i)33-s + ⋯
L(s)  = 1  + (0.999 − 0.999i)3-s + (0.447 − 0.894i)5-s + (0.654 + 0.654i)7-s − 0.999i·9-s + 1.04i·11-s + (−0.277 − 0.277i)13-s + (−0.447 − 1.34i)15-s + (0.242 − 0.242i)17-s − 1.58·19-s + 1.30·21-s + (−0.361 + 0.361i)23-s + (−0.600 − 0.800i)25-s − 0.742i·29-s + 0.622i·31-s + (1.04 + 1.04i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68424 - 0.939028i\)
\(L(\frac12)\) \(\approx\) \(1.68424 - 0.939028i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good3 \( 1 + (-1.73 + 1.73i)T - 3iT^{2} \)
7 \( 1 + (-1.73 - 1.73i)T + 7iT^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 + (1.73 - 1.73i)T - 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + (-1.73 + 1.73i)T - 43iT^{2} \)
47 \( 1 + (-1.73 - 1.73i)T + 47iT^{2} \)
53 \( 1 + (-7 - 7i)T + 53iT^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-5.19 - 5.19i)T + 67iT^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (7 + 7i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (12.1 - 12.1i)T - 83iT^{2} \)
89 \( 1 - 8iT - 89T^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−11.97255529953149342334579206648, −10.39051291657609199293076744630, −9.312110661948539489251710584979, −8.520030397748944726355109838589, −7.905255167452555018665217257026, −6.82308850182998501342410649284, −5.51534163991162366790102186721, −4.39256177429860271760739604785, −2.46614160490971712677917911034, −1.66652469312998831091684148606, 2.27341516007885997535894760599, 3.51405606391579631897399383262, 4.37068983109001058438863996460, 5.85169404000169724182152534028, 7.06817466296699457556799177892, 8.242490332142431549097990350949, 8.939767614748336886184171536571, 10.09284516516543665875676102064, 10.61451685782334521210393634844, 11.37936966706013061813212435880

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