Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.883 - 0.468i$
Motivic weight 5
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

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.883 - 0.468i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.883 - 0.468i)\)
\(L(3)\)  \(\approx\)  \(4.836181871\)
\(L(\frac12)\)  \(\approx\)  \(4.836181871\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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