Properties

Label 2-80-80.37-c10-0-102
Degree $2$
Conductor $80$
Sign $-0.720 + 0.693i$
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  + (−31.1 − 7.44i)2-s − 73.3i·3-s + (913. + 463. i)4-s + (3.12e3 + 79.4i)5-s + (−546. + 2.28e3i)6-s + (1.91e4 − 1.91e4i)7-s + (−2.49e4 − 2.12e4i)8-s + 5.36e4·9-s + (−9.66e4 − 2.57e4i)10-s + (−1.01e5 − 1.01e5i)11-s + (3.39e4 − 6.69e4i)12-s − 4.59e5i·13-s + (−7.36e5 + 4.52e5i)14-s + (5.83e3 − 2.29e5i)15-s + (6.19e5 + 8.46e5i)16-s + (−1.57e6 − 1.57e6i)17-s + ⋯
L(s)  = 1  + (−0.972 − 0.232i)2-s − 0.301i·3-s + (0.891 + 0.452i)4-s + (0.999 + 0.0254i)5-s + (−0.0702 + 0.293i)6-s + (1.13 − 1.13i)7-s + (−0.761 − 0.647i)8-s + 0.908·9-s + (−0.966 − 0.257i)10-s + (−0.632 − 0.632i)11-s + (0.136 − 0.269i)12-s − 1.23i·13-s + (−1.37 + 0.841i)14-s + (0.00767 − 0.301i)15-s + (0.590 + 0.807i)16-s + (−1.10 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.720 + 0.693i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ -0.720 + 0.693i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.631416 - 1.56496i\)
\(L(\frac12)\) \(\approx\) \(0.631416 - 1.56496i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (31.1 + 7.44i)T \)
5 \( 1 + (-3.12e3 - 79.4i)T \)
good3 \( 1 + 73.3iT - 5.90e4T^{2} \)
7 \( 1 + (-1.91e4 + 1.91e4i)T - 2.82e8iT^{2} \)
11 \( 1 + (1.01e5 + 1.01e5i)T + 2.59e10iT^{2} \)
13 \( 1 + 4.59e5iT - 1.37e11T^{2} \)
17 \( 1 + (1.57e6 + 1.57e6i)T + 2.01e12iT^{2} \)
19 \( 1 + (-5.83e5 - 5.83e5i)T + 6.13e12iT^{2} \)
23 \( 1 + (4.60e6 + 4.60e6i)T + 4.14e13iT^{2} \)
29 \( 1 + (4.75e6 + 4.75e6i)T + 4.20e14iT^{2} \)
31 \( 1 - 3.46e7T + 8.19e14T^{2} \)
37 \( 1 - 1.05e8iT - 4.80e15T^{2} \)
41 \( 1 - 1.27e8iT - 1.34e16T^{2} \)
43 \( 1 + 1.79e8T + 2.16e16T^{2} \)
47 \( 1 + (-2.82e7 - 2.82e7i)T + 5.25e16iT^{2} \)
53 \( 1 - 4.31e7T + 1.74e17T^{2} \)
59 \( 1 + (-2.15e8 + 2.15e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (8.91e8 - 8.91e8i)T - 7.13e17iT^{2} \)
67 \( 1 + 5.32e8T + 1.82e18T^{2} \)
71 \( 1 - 4.99e8iT - 3.25e18T^{2} \)
73 \( 1 + (8.77e8 + 8.77e8i)T + 4.29e18iT^{2} \)
79 \( 1 + 1.04e9iT - 9.46e18T^{2} \)
83 \( 1 + 1.28e9iT - 1.55e19T^{2} \)
89 \( 1 + 1.06e10T + 3.11e19T^{2} \)
97 \( 1 + (-4.24e9 - 4.24e9i)T + 7.37e19iT^{2} \)
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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

−11.59375068113748849556570066157, −10.47189053103872293433451744284, −10.00935643069965872598588915414, −8.390965583032391155835895850450, −7.55940779675796567742506875118, −6.41034831166489811600396980787, −4.73945628084385688299187216028, −2.76946083748228001310838218821, −1.47684432867360526988183075165, −0.59762855567559374965176045821, 1.75830983619181745947958321586, 2.06699125537965598943287929209, 4.68955322805569419986440274127, 5.86124804549796824990614899646, 7.11157791586140182919811095358, 8.509511307703841667710780536126, 9.336534324338956241071810240522, 10.29866267463757627705563065536, 11.34925971819987457118899234522, 12.53634750086510132924440167118

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