Properties

Label 2-80-80.13-c10-0-96
Degree $2$
Conductor $80$
Sign $-0.140 + 0.990i$
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  + (−29.0 + 13.4i)2-s + 161. i·3-s + (660. − 782. i)4-s + (462. − 3.09e3i)5-s + (−2.17e3 − 4.69e3i)6-s + (−1.32e4 − 1.32e4i)7-s + (−8.63e3 + 3.16e4i)8-s + 3.29e4·9-s + (2.82e4 + 9.59e4i)10-s + (1.45e5 − 1.45e5i)11-s + (1.26e5 + 1.06e5i)12-s + 9.30e4i·13-s + (5.64e5 + 2.06e5i)14-s + (4.99e5 + 7.46e4i)15-s + (−1.75e5 − 1.03e6i)16-s + (4.81e5 − 4.81e5i)17-s + ⋯
L(s)  = 1  + (−0.906 + 0.421i)2-s + 0.665i·3-s + (0.645 − 0.763i)4-s + (0.147 − 0.989i)5-s + (−0.280 − 0.603i)6-s + (−0.790 − 0.790i)7-s + (−0.263 + 0.964i)8-s + 0.557·9-s + (0.282 + 0.959i)10-s + (0.906 − 0.906i)11-s + (0.508 + 0.429i)12-s + 0.250i·13-s + (1.04 + 0.384i)14-s + (0.657 + 0.0983i)15-s + (−0.167 − 0.985i)16-s + (0.338 − 0.338i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.140 + 0.990i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ -0.140 + 0.990i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.671983 - 0.774261i\)
\(L(\frac12)\) \(\approx\) \(0.671983 - 0.774261i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (29.0 - 13.4i)T \)
5 \( 1 + (-462. + 3.09e3i)T \)
good3 \( 1 - 161. iT - 5.90e4T^{2} \)
7 \( 1 + (1.32e4 + 1.32e4i)T + 2.82e8iT^{2} \)
11 \( 1 + (-1.45e5 + 1.45e5i)T - 2.59e10iT^{2} \)
13 \( 1 - 9.30e4iT - 1.37e11T^{2} \)
17 \( 1 + (-4.81e5 + 4.81e5i)T - 2.01e12iT^{2} \)
19 \( 1 + (-1.62e6 + 1.62e6i)T - 6.13e12iT^{2} \)
23 \( 1 + (4.96e5 - 4.96e5i)T - 4.14e13iT^{2} \)
29 \( 1 + (1.67e7 - 1.67e7i)T - 4.20e14iT^{2} \)
31 \( 1 - 2.35e7T + 8.19e14T^{2} \)
37 \( 1 + 8.53e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.28e7iT - 1.34e16T^{2} \)
43 \( 1 + 9.53e7T + 2.16e16T^{2} \)
47 \( 1 + (1.87e8 - 1.87e8i)T - 5.25e16iT^{2} \)
53 \( 1 - 6.79e6T + 1.74e17T^{2} \)
59 \( 1 + (1.99e7 + 1.99e7i)T + 5.11e17iT^{2} \)
61 \( 1 + (-1.02e8 - 1.02e8i)T + 7.13e17iT^{2} \)
67 \( 1 + 1.06e9T + 1.82e18T^{2} \)
71 \( 1 + 2.56e9iT - 3.25e18T^{2} \)
73 \( 1 + (-2.17e9 + 2.17e9i)T - 4.29e18iT^{2} \)
79 \( 1 - 1.19e8iT - 9.46e18T^{2} \)
83 \( 1 + 4.47e9iT - 1.55e19T^{2} \)
89 \( 1 - 8.10e9T + 3.11e19T^{2} \)
97 \( 1 + (-3.22e9 + 3.22e9i)T - 7.37e19iT^{2} \)
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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

−11.76165525016261368503547254690, −10.56135524973677906519248252912, −9.504850498560094241654657550625, −9.033206667350689503502095185661, −7.52457509674445268878931383577, −6.34836954736227136819175727334, −4.96995944962137973226723058231, −3.55964345819000963619115566321, −1.36063055705018973641800142883, −0.40214932772154020592385831147, 1.36372019773750637170095770342, 2.41755059621461297365051150033, 3.66143919087913836529111879765, 6.22988143044166673913093981844, 6.96171017763889921212415491596, 8.034315224967616414385437086511, 9.634643904225241191886891661687, 10.06710584427668177103184145497, 11.66164650322876659922564363713, 12.31302166465327373311264859527

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