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Properties

Label 2-80-16.11-c10-0-50
Degree $2$
Conductor $80$
Sign $-0.893 - 0.450i$
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  + (26.2 + 18.2i)2-s + (328. + 328. i)3-s + (356. + 959. i)4-s + (−988. − 988. i)5-s + (2.63e3 + 1.46e4i)6-s + 2.19e4·7-s + (−8.16e3 + 3.17e4i)8-s + 1.57e5i·9-s + (−7.91e3 − 4.40e4i)10-s + (5.63e4 − 5.63e4i)11-s + (−1.98e5 + 4.32e5i)12-s + (−2.05e5 + 2.05e5i)13-s + (5.77e5 + 4.01e5i)14-s − 6.49e5i·15-s + (−7.94e5 + 6.84e5i)16-s + 1.60e6·17-s + ⋯
L(s)  = 1  + (0.821 + 0.570i)2-s + (1.35 + 1.35i)3-s + (0.348 + 0.937i)4-s + (−0.316 − 0.316i)5-s + (0.338 + 1.88i)6-s + 1.30·7-s + (−0.249 + 0.968i)8-s + 2.66i·9-s + (−0.0791 − 0.440i)10-s + (0.350 − 0.350i)11-s + (−0.797 + 1.73i)12-s + (−0.554 + 0.554i)13-s + (1.07 + 0.747i)14-s − 0.855i·15-s + (−0.757 + 0.652i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.450i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.893 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.893 - 0.450i$
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: \((2,\ 80,\ (\ :5),\ -0.893 - 0.450i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.38270 + 5.81588i\)
\(L(\frac12)\) \(\approx\) \(1.38270 + 5.81588i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-26.2 - 18.2i)T \)
5 \( 1 + (988. + 988. i)T \)
good3 \( 1 + (-328. - 328. i)T + 5.90e4iT^{2} \)
7 \( 1 - 2.19e4T + 2.82e8T^{2} \)
11 \( 1 + (-5.63e4 + 5.63e4i)T - 2.59e10iT^{2} \)
13 \( 1 + (2.05e5 - 2.05e5i)T - 1.37e11iT^{2} \)
17 \( 1 - 1.60e6T + 2.01e12T^{2} \)
19 \( 1 + (3.24e6 + 3.24e6i)T + 6.13e12iT^{2} \)
23 \( 1 - 5.82e6T + 4.14e13T^{2} \)
29 \( 1 + (3.18e6 - 3.18e6i)T - 4.20e14iT^{2} \)
31 \( 1 - 2.40e7iT - 8.19e14T^{2} \)
37 \( 1 + (-3.44e7 - 3.44e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 4.87e7iT - 1.34e16T^{2} \)
43 \( 1 + (-9.50e7 + 9.50e7i)T - 2.16e16iT^{2} \)
47 \( 1 + 4.24e8iT - 5.25e16T^{2} \)
53 \( 1 + (2.78e8 + 2.78e8i)T + 1.74e17iT^{2} \)
59 \( 1 + (-4.05e8 + 4.05e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (8.25e8 - 8.25e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (-9.69e8 - 9.69e8i)T + 1.82e18iT^{2} \)
71 \( 1 + 6.58e8T + 3.25e18T^{2} \)
73 \( 1 + 2.67e9iT - 4.29e18T^{2} \)
79 \( 1 + 1.96e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.37e9 + 1.37e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 2.40e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.32e10T + 7.37e19T^{2} \)
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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

−13.22910694201024418192688913380, −11.67574534956854926890913629024, −10.65510111605040279144951705479, −8.984892561773778075233130534637, −8.416115568936797696484371242886, −7.33344906904000869125117743646, −5.07967775957202712609113518015, −4.51969708047666848213416745268, −3.41489475271208313797433193287, −2.13918398628817827990742045792, 1.04062074233348666675716064172, 1.92379148604977059471421812550, 2.96779556484078983128509676159, 4.23719725485011548447753396848, 6.05500400130918286466939576316, 7.44227447767240421867198971648, 8.114474139665764852787140531493, 9.610591224249483093782691715540, 11.12481716251736898384152611747, 12.31154056635078162911185037994

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