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Properties

Label 2-80-5.2-c8-0-9
Degree $2$
Conductor $80$
Sign $0.524 - 0.851i$
Analytic cond. $32.5902$
Root an. cond. $5.70879$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (20.3 − 20.3i)3-s + (558. − 280. i)5-s + (2.41e3 + 2.41e3i)7-s + 5.73e3i·9-s + 981.·11-s + (−2.65e4 + 2.65e4i)13-s + (5.65e3 − 1.70e4i)15-s + (−1.85e4 − 1.85e4i)17-s + 5.03e4i·19-s + 9.82e4·21-s + (−1.36e4 + 1.36e4i)23-s + (2.33e5 − 3.13e5i)25-s + (2.49e5 + 2.49e5i)27-s + 1.05e6i·29-s − 1.09e6·31-s + ⋯
L(s)  = 1  + (0.251 − 0.251i)3-s + (0.893 − 0.448i)5-s + (1.00 + 1.00i)7-s + 0.873i·9-s + 0.0670·11-s + (−0.930 + 0.930i)13-s + (0.111 − 0.336i)15-s + (−0.221 − 0.221i)17-s + 0.386i·19-s + 0.504·21-s + (−0.0487 + 0.0487i)23-s + (0.597 − 0.802i)25-s + (0.470 + 0.470i)27-s + 1.48i·29-s − 1.18·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.524 - 0.851i$
Analytic conductor: \(32.5902\)
Root analytic conductor: \(5.70879\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :4),\ 0.524 - 0.851i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.25590 + 1.26025i\)
\(L(\frac12)\) \(\approx\) \(2.25590 + 1.26025i\)
\(L(5)\) 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 \( 1 + (-558. + 280. i)T \)
good3 \( 1 + (-20.3 + 20.3i)T - 6.56e3iT^{2} \)
7 \( 1 + (-2.41e3 - 2.41e3i)T + 5.76e6iT^{2} \)
11 \( 1 - 981.T + 2.14e8T^{2} \)
13 \( 1 + (2.65e4 - 2.65e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (1.85e4 + 1.85e4i)T + 6.97e9iT^{2} \)
19 \( 1 - 5.03e4iT - 1.69e10T^{2} \)
23 \( 1 + (1.36e4 - 1.36e4i)T - 7.83e10iT^{2} \)
29 \( 1 - 1.05e6iT - 5.00e11T^{2} \)
31 \( 1 + 1.09e6T + 8.52e11T^{2} \)
37 \( 1 + (-7.78e4 - 7.78e4i)T + 3.51e12iT^{2} \)
41 \( 1 - 5.54e5T + 7.98e12T^{2} \)
43 \( 1 + (-1.07e6 + 1.07e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (-4.06e6 - 4.06e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (-1.88e6 + 1.88e6i)T - 6.22e13iT^{2} \)
59 \( 1 - 1.27e7iT - 1.46e14T^{2} \)
61 \( 1 - 1.40e7T + 1.91e14T^{2} \)
67 \( 1 + (-9.54e6 - 9.54e6i)T + 4.06e14iT^{2} \)
71 \( 1 - 2.82e7T + 6.45e14T^{2} \)
73 \( 1 + (-1.11e7 + 1.11e7i)T - 8.06e14iT^{2} \)
79 \( 1 + 6.87e7iT - 1.51e15T^{2} \)
83 \( 1 + (-3.29e6 + 3.29e6i)T - 2.25e15iT^{2} \)
89 \( 1 + 7.97e7iT - 3.93e15T^{2} \)
97 \( 1 + (1.96e7 + 1.96e7i)T + 7.83e15iT^{2} \)
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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

−12.90644165003965772925828815540, −11.95165527425956047118663494316, −10.74240828919795938586986426795, −9.332044883445238499003059886705, −8.537831952170505956491365388009, −7.23436443250786967207080999987, −5.58036844153989171747917861314, −4.75361999347381270000117281675, −2.36690215774053168695456026911, −1.64447890066465045683026540412, 0.75903069251024380530179147078, 2.34828303579974266311423715070, 3.90876675957423672405051923004, 5.32918828488851531371665830087, 6.77490976023050498516959520474, 7.926698718808688668382134245318, 9.398648236065585227322708702699, 10.28455874281964014960742979342, 11.25012327513610188691017152490, 12.69056170781075630243617404194

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