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Properties

Label 2-80-5.4-c19-0-36
Degree $2$
Conductor $80$
Sign $0.792 + 0.609i$
Analytic cond. $183.053$
Root an. cond. $13.5297$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.08e3i·3-s + (3.46e6 + 2.66e6i)5-s − 8.07e7i·7-s + 1.11e9·9-s − 1.18e10·11-s + 5.69e10i·13-s + (−1.88e10 + 2.45e10i)15-s − 6.43e11i·17-s − 2.02e12·19-s + 5.72e11·21-s − 4.89e12i·23-s + (4.91e12 + 1.84e13i)25-s + 1.61e13i·27-s + 8.54e13·29-s + 1.78e14·31-s + ⋯
L(s)  = 1  + 0.207i·3-s + (0.792 + 0.609i)5-s − 0.756i·7-s + 0.956·9-s − 1.51·11-s + 1.48i·13-s + (−0.126 + 0.164i)15-s − 1.31i·17-s − 1.44·19-s + 0.157·21-s − 0.566i·23-s + (0.257 + 0.966i)25-s + 0.406i·27-s + 1.09·29-s + 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.792 + 0.609i$
Analytic conductor: \(183.053\)
Root analytic conductor: \(13.5297\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :19/2),\ 0.792 + 0.609i)\)

Particular Values

\(L(10)\) \(\approx\) \(2.162297860\)
\(L(\frac12)\) \(\approx\) \(2.162297860\)
\(L(\frac{21}{2})\) 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 \( 1 + (-3.46e6 - 2.66e6i)T \)
good3 \( 1 - 7.08e3iT - 1.16e9T^{2} \)
7 \( 1 + 8.07e7iT - 1.13e16T^{2} \)
11 \( 1 + 1.18e10T + 6.11e19T^{2} \)
13 \( 1 - 5.69e10iT - 1.46e21T^{2} \)
17 \( 1 + 6.43e11iT - 2.39e23T^{2} \)
19 \( 1 + 2.02e12T + 1.97e24T^{2} \)
23 \( 1 + 4.89e12iT - 7.46e25T^{2} \)
29 \( 1 - 8.54e13T + 6.10e27T^{2} \)
31 \( 1 - 1.78e14T + 2.16e28T^{2} \)
37 \( 1 + 4.42e14iT - 6.24e29T^{2} \)
41 \( 1 + 2.92e15T + 4.39e30T^{2} \)
43 \( 1 - 1.00e15iT - 1.08e31T^{2} \)
47 \( 1 + 7.76e15iT - 5.88e31T^{2} \)
53 \( 1 + 4.30e16iT - 5.77e32T^{2} \)
59 \( 1 - 7.84e16T + 4.42e33T^{2} \)
61 \( 1 - 4.14e16T + 8.34e33T^{2} \)
67 \( 1 - 3.97e17iT - 4.95e34T^{2} \)
71 \( 1 + 4.02e17T + 1.49e35T^{2} \)
73 \( 1 - 6.89e16iT - 2.53e35T^{2} \)
79 \( 1 + 1.39e17T + 1.13e36T^{2} \)
83 \( 1 + 3.15e18iT - 2.90e36T^{2} \)
89 \( 1 + 5.27e18T + 1.09e37T^{2} \)
97 \( 1 - 8.01e18iT - 5.60e37T^{2} \)
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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

−10.27349096985939298149432365996, −10.08528104882202128363138171031, −8.615150706311135121049604842897, −7.14256382185928738097871429384, −6.62120083727385013555930413655, −5.04481062199315747336191919259, −4.19425170882263808554805860631, −2.70826496149821832671770642144, −1.86003945239799476227627778331, −0.45182386064966140312373965961, 0.861106391840957874300577920581, 1.95027856551970980944781145699, 2.84667079484630573082351769957, 4.53104577116850496260598902241, 5.49738856448036405577566162794, 6.34434326202579800042375056775, 7.959699572159130478607777775195, 8.567310965947984019466560531106, 10.13443322398890915325188577500, 10.46502448805680229948227803383

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