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Properties

Label 2-320-40.29-c5-0-45
Degree $2$
Conductor $320$
Sign $0.883 + 0.468i$
Analytic cond. $51.3228$
Root an. cond. $7.16399$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 24.8·3-s + (53.4 − 16.3i)5-s − 100. i·7-s + 372.·9-s + 5.36i·11-s + 506.·13-s + (1.32e3 − 406. i)15-s + 1.45e3i·17-s − 722. i·19-s − 2.48e3i·21-s + 3.08e3i·23-s + (2.58e3 − 1.75e3i)25-s + 3.20e3·27-s − 7.29e3i·29-s + 8.60e3·31-s + ⋯
L(s)  = 1  + 1.59·3-s + (0.956 − 0.293i)5-s − 0.773i·7-s + 1.53·9-s + 0.0133i·11-s + 0.831·13-s + (1.52 − 0.466i)15-s + 1.22i·17-s − 0.459i·19-s − 1.23i·21-s + 1.21i·23-s + (0.828 − 0.560i)25-s + 0.846·27-s − 1.61i·29-s + 1.60·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.883 + 0.468i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ 0.883 + 0.468i)\)

Particular Values

\(L(3)\) \(\approx\) \(4.836181871\)
\(L(\frac12)\) \(\approx\) \(4.836181871\)
\(L(\frac{7}{2})\) 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 \( 1 + (-53.4 + 16.3i)T \)
good3 \( 1 - 24.8T + 243T^{2} \)
7 \( 1 + 100. iT - 1.68e4T^{2} \)
11 \( 1 - 5.36iT - 1.61e5T^{2} \)
13 \( 1 - 506.T + 3.71e5T^{2} \)
17 \( 1 - 1.45e3iT - 1.41e6T^{2} \)
19 \( 1 + 722. iT - 2.47e6T^{2} \)
23 \( 1 - 3.08e3iT - 6.43e6T^{2} \)
29 \( 1 + 7.29e3iT - 2.05e7T^{2} \)
31 \( 1 - 8.60e3T + 2.86e7T^{2} \)
37 \( 1 - 5.27e3T + 6.93e7T^{2} \)
41 \( 1 + 1.81e4T + 1.15e8T^{2} \)
43 \( 1 + 8.34e3T + 1.47e8T^{2} \)
47 \( 1 + 2.21e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.33e4T + 4.18e8T^{2} \)
59 \( 1 + 1.09e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.09e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.30e4T + 1.35e9T^{2} \)
71 \( 1 + 3.60e4T + 1.80e9T^{2} \)
73 \( 1 - 6.42e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.86e4T + 3.07e9T^{2} \)
83 \( 1 + 4.97e4T + 3.93e9T^{2} \)
89 \( 1 - 7.76e3T + 5.58e9T^{2} \)
97 \( 1 + 3.68e4iT - 8.58e9T^{2} \)
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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

−10.23621419211644465148245149344, −9.834171491962772632488406659155, −8.681185908552781836411423513879, −8.204387450743481821083097827118, −7.01286706990500084634184406308, −5.89516798009084789732285391809, −4.34652991360854097358603528776, −3.40618519735165605008838352384, −2.16858178137781383336723912090, −1.17691565364933442625582831168, 1.43517707371419276178276038221, 2.57445262122121008970094693007, 3.18950306308672897790761448851, 4.76438105848953471521771689218, 6.08468022327579487119927944120, 7.09999787861680475913590824861, 8.389827035283501887841790206127, 8.879177801127202197800157496946, 9.719477008450687198509695516365, 10.56060620751210999508836290478

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