Properties

Label 2-80-5.2-c8-0-22
Degree $2$
Conductor $80$
Sign $-0.969 - 0.246i$
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  + (90.9 − 90.9i)3-s + (−434. − 449. i)5-s + (−508. − 508. i)7-s − 9.98e3i·9-s + 7.02e3·11-s + (−8.07e3 + 8.07e3i)13-s + (−8.03e4 − 1.36e3i)15-s + (−1.02e5 − 1.02e5i)17-s − 5.95e4i·19-s − 9.24e4·21-s + (−1.32e5 + 1.32e5i)23-s + (−1.32e4 + 3.90e5i)25-s + (−3.11e5 − 3.11e5i)27-s + 3.92e5i·29-s + 5.07e5·31-s + ⋯
L(s)  = 1  + (1.12 − 1.12i)3-s + (−0.694 − 0.719i)5-s + (−0.211 − 0.211i)7-s − 1.52i·9-s + 0.479·11-s + (−0.282 + 0.282i)13-s + (−1.58 − 0.0269i)15-s + (−1.23 − 1.23i)17-s − 0.457i·19-s − 0.475·21-s + (−0.474 + 0.474i)23-s + (−0.0339 + 0.999i)25-s + (−0.586 − 0.586i)27-s + 0.554i·29-s + 0.549·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.969 - 0.246i$
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: \((2,\ 80,\ (\ :4),\ -0.969 - 0.246i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.181027 + 1.44766i\)
\(L(\frac12)\) \(\approx\) \(0.181027 + 1.44766i\)
\(L(5)\) 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 \( 1 + (434. + 449. i)T \)
good3 \( 1 + (-90.9 + 90.9i)T - 6.56e3iT^{2} \)
7 \( 1 + (508. + 508. i)T + 5.76e6iT^{2} \)
11 \( 1 - 7.02e3T + 2.14e8T^{2} \)
13 \( 1 + (8.07e3 - 8.07e3i)T - 8.15e8iT^{2} \)
17 \( 1 + (1.02e5 + 1.02e5i)T + 6.97e9iT^{2} \)
19 \( 1 + 5.95e4iT - 1.69e10T^{2} \)
23 \( 1 + (1.32e5 - 1.32e5i)T - 7.83e10iT^{2} \)
29 \( 1 - 3.92e5iT - 5.00e11T^{2} \)
31 \( 1 - 5.07e5T + 8.52e11T^{2} \)
37 \( 1 + (6.10e4 + 6.10e4i)T + 3.51e12iT^{2} \)
41 \( 1 + 1.81e6T + 7.98e12T^{2} \)
43 \( 1 + (1.47e6 - 1.47e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (-1.79e6 - 1.79e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (5.66e6 - 5.66e6i)T - 6.22e13iT^{2} \)
59 \( 1 + 1.74e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.96e7T + 1.91e14T^{2} \)
67 \( 1 + (-1.12e7 - 1.12e7i)T + 4.06e14iT^{2} \)
71 \( 1 + 3.01e7T + 6.45e14T^{2} \)
73 \( 1 + (-2.52e7 + 2.52e7i)T - 8.06e14iT^{2} \)
79 \( 1 + 8.14e6iT - 1.51e15T^{2} \)
83 \( 1 + (-1.99e7 + 1.99e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 8.20e7iT - 3.93e15T^{2} \)
97 \( 1 + (-3.37e7 - 3.37e7i)T + 7.83e15iT^{2} \)
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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

−12.34236556361344313449856633634, −11.42206632040376959549955648596, −9.409622658392384234958654466913, −8.652245134459299541960158929867, −7.55674193149675283150825002116, −6.71586389469267682971193455666, −4.61358367406642615657577747576, −3.14659660491641450849390886043, −1.72010963139522757146327657095, −0.36522932956607404120631907800, 2.36154472389570283169808709543, 3.56634750379456692411125756464, 4.40127653075212226832920838329, 6.40735563018568854949403902524, 7.962638503015045290129953618222, 8.813530310012713106290313253374, 10.01818010258391302866894969250, 10.81969327720687606076751367903, 12.16773951357810367538948246359, 13.64015559440327403292320052705

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