Properties

Degree 2
Conductor $ 2^{4} \cdot 5 $
Sign $0.814 + 0.580i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 37.6i·3-s + (1.13e3 + 810. i)5-s − 5.31e3i·7-s + 1.82e4·9-s − 1.04e4·11-s − 7.96e4i·13-s + (−3.05e4 + 4.28e4i)15-s − 3.13e5i·17-s − 2.46e5·19-s + 2.00e5·21-s − 7.21e5i·23-s + (6.38e5 + 1.84e6i)25-s + 1.42e6i·27-s − 2.56e6·29-s + 3.29e6·31-s + ⋯
L(s)  = 1  + 0.268i·3-s + (0.814 + 0.580i)5-s − 0.836i·7-s + 0.928·9-s − 0.214·11-s − 0.773i·13-s + (−0.155 + 0.218i)15-s − 0.911i·17-s − 0.434·19-s + 0.224·21-s − 0.537i·23-s + (0.326 + 0.945i)25-s + 0.517i·27-s − 0.674·29-s + 0.640·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(80\)    =    \(2^{4} \cdot 5\)
\( \varepsilon \)  =  $0.814 + 0.580i$
motivic weight  =  \(9\)
character  :  $\chi_{80} (49, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 80,\ (\ :9/2),\ 0.814 + 0.580i)$
$L(5)$  $\approx$  $2.30954 - 0.738464i$
$L(\frac12)$  $\approx$  $2.30954 - 0.738464i$
$L(\frac{11}{2})$   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 \( 1 + (-1.13e3 - 810. i)T \)
good3 \( 1 - 37.6iT - 1.96e4T^{2} \)
7 \( 1 + 5.31e3iT - 4.03e7T^{2} \)
11 \( 1 + 1.04e4T + 2.35e9T^{2} \)
13 \( 1 + 7.96e4iT - 1.06e10T^{2} \)
17 \( 1 + 3.13e5iT - 1.18e11T^{2} \)
19 \( 1 + 2.46e5T + 3.22e11T^{2} \)
23 \( 1 + 7.21e5iT - 1.80e12T^{2} \)
29 \( 1 + 2.56e6T + 1.45e13T^{2} \)
31 \( 1 - 3.29e6T + 2.64e13T^{2} \)
37 \( 1 + 1.40e7iT - 1.29e14T^{2} \)
41 \( 1 - 1.70e7T + 3.27e14T^{2} \)
43 \( 1 + 2.92e7iT - 5.02e14T^{2} \)
47 \( 1 - 4.10e7iT - 1.11e15T^{2} \)
53 \( 1 + 5.67e7iT - 3.29e15T^{2} \)
59 \( 1 - 1.60e8T + 8.66e15T^{2} \)
61 \( 1 - 5.33e7T + 1.16e16T^{2} \)
67 \( 1 + 2.80e8iT - 2.72e16T^{2} \)
71 \( 1 - 8.97e7T + 4.58e16T^{2} \)
73 \( 1 + 7.60e7iT - 5.88e16T^{2} \)
79 \( 1 - 4.10e8T + 1.19e17T^{2} \)
83 \( 1 - 5.21e8iT - 1.86e17T^{2} \)
89 \( 1 + 2.37e8T + 3.50e17T^{2} \)
97 \( 1 - 6.03e8iT - 7.60e17T^{2} \)
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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

−12.64951880890264685117951504154, −10.93806728192155583728722770356, −10.28372347079649882035797889517, −9.358065627294602490233118025099, −7.63861570945178632431466401726, −6.68369588665202978028425921971, −5.24430808831981943073295864868, −3.84334344800049004983379725963, −2.34224561462314009350880672692, −0.74255264177482963577150369892, 1.29040815091873406371963258123, 2.28692208723966265863125018409, 4.28525216618922610783559680238, 5.61154466014508570380407418148, 6.68872787780404691940798313582, 8.232749292513288873417539740518, 9.294172940264578056886199409622, 10.23929730552541806673572829256, 11.75273604533318181809747496683, 12.77723665561226988901092530920

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