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Properties

Label 2-80-16.11-c10-0-47
Degree $2$
Conductor $80$
Sign $0.900 - 0.434i$
Analytic cond. $50.8285$
Root an. cond. $7.12941$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (29.7 + 11.8i)2-s + (−233. − 233. i)3-s + (744. + 703. i)4-s + (988. + 988. i)5-s + (−4.18e3 − 9.70e3i)6-s + 3.14e4·7-s + (1.38e4 + 2.97e4i)8-s + 4.99e4i·9-s + (1.76e4 + 4.10e4i)10-s + (1.07e5 − 1.07e5i)11-s + (−9.59e3 − 3.37e5i)12-s + (−3.28e5 + 3.28e5i)13-s + (9.35e5 + 3.72e5i)14-s − 4.61e5i·15-s + (5.95e4 + 1.04e6i)16-s − 1.87e6·17-s + ⋯
L(s)  = 1  + (0.929 + 0.369i)2-s + (−0.960 − 0.960i)3-s + (0.726 + 0.686i)4-s + (0.316 + 0.316i)5-s + (−0.537 − 1.24i)6-s + 1.87·7-s + (0.421 + 0.906i)8-s + 0.845i·9-s + (0.176 + 0.410i)10-s + (0.666 − 0.666i)11-s + (−0.0385 − 1.35i)12-s + (−0.884 + 0.884i)13-s + (1.73 + 0.691i)14-s − 0.607i·15-s + (0.0567 + 0.998i)16-s − 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.900 - 0.434i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ 0.900 - 0.434i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(3.62041 + 0.827645i\)
\(L(\frac12)\) \(\approx\) \(3.62041 + 0.827645i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-29.7 - 11.8i)T \)
5 \( 1 + (-988. - 988. i)T \)
good3 \( 1 + (233. + 233. i)T + 5.90e4iT^{2} \)
7 \( 1 - 3.14e4T + 2.82e8T^{2} \)
11 \( 1 + (-1.07e5 + 1.07e5i)T - 2.59e10iT^{2} \)
13 \( 1 + (3.28e5 - 3.28e5i)T - 1.37e11iT^{2} \)
17 \( 1 + 1.87e6T + 2.01e12T^{2} \)
19 \( 1 + (-8.45e5 - 8.45e5i)T + 6.13e12iT^{2} \)
23 \( 1 - 7.20e6T + 4.14e13T^{2} \)
29 \( 1 + (-1.45e7 + 1.45e7i)T - 4.20e14iT^{2} \)
31 \( 1 - 3.04e7iT - 8.19e14T^{2} \)
37 \( 1 + (-6.02e7 - 6.02e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 1.08e7iT - 1.34e16T^{2} \)
43 \( 1 + (-1.77e8 + 1.77e8i)T - 2.16e16iT^{2} \)
47 \( 1 + 1.29e8iT - 5.25e16T^{2} \)
53 \( 1 + (-1.56e8 - 1.56e8i)T + 1.74e17iT^{2} \)
59 \( 1 + (-4.70e7 + 4.70e7i)T - 5.11e17iT^{2} \)
61 \( 1 + (-6.34e8 + 6.34e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (1.02e9 + 1.02e9i)T + 1.82e18iT^{2} \)
71 \( 1 + 2.14e9T + 3.25e18T^{2} \)
73 \( 1 - 3.46e9iT - 4.29e18T^{2} \)
79 \( 1 - 2.44e9iT - 9.46e18T^{2} \)
83 \( 1 + (-3.28e9 - 3.28e9i)T + 1.55e19iT^{2} \)
89 \( 1 - 2.04e9iT - 3.11e19T^{2} \)
97 \( 1 - 2.50e9T + 7.37e19T^{2} \)
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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

−12.25299522534583460506762996320, −11.51229925589293503914897008822, −10.99643718291961956924499419039, −8.626910157002783754217406641416, −7.31520302095072860762524558898, −6.56678732718931981864030589718, −5.38263154955234152474524869073, −4.42150705357397204768553719322, −2.27765963938203615207764314824, −1.23072822605347058363633305167, 0.930580692426330315430845603311, 2.30591856855264839625956389545, 4.52841894639212265282795218530, 4.71106574229912871412099981315, 5.78841092528267893075729268251, 7.41108577099724087253030569972, 9.265945105432879448584136082200, 10.53665736410215194366667736306, 11.21013325535080770502313923441, 11.97859869850886067694957744625

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