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Properties

Label 2-80-80.29-c5-0-40
Degree $2$
Conductor $80$
Sign $-0.452 + 0.892i$
Analytic cond. $12.8307$
Root an. cond. $3.58199$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.15 + 2.32i)2-s + (−9.83 + 9.83i)3-s + (21.2 − 23.9i)4-s + (46.4 − 31.0i)5-s + (27.9 − 73.5i)6-s − 45.7·7-s + (−53.9 + 172. i)8-s + 49.4i·9-s + (−167. + 268. i)10-s + (−266. + 266. i)11-s + (26.7 + 444. i)12-s + (−19.5 + 19.5i)13-s + (235. − 106. i)14-s + (−151. + 762. i)15-s + (−122. − 1.01e3i)16-s − 891. i·17-s + ⋯
L(s)  = 1  + (−0.911 + 0.410i)2-s + (−0.631 + 0.631i)3-s + (0.663 − 0.748i)4-s + (0.831 − 0.555i)5-s + (0.316 − 0.834i)6-s − 0.352·7-s + (−0.297 + 0.954i)8-s + 0.203i·9-s + (−0.530 + 0.847i)10-s + (−0.664 + 0.664i)11-s + (0.0536 + 0.890i)12-s + (−0.0321 + 0.0321i)13-s + (0.321 − 0.144i)14-s + (−0.174 + 0.875i)15-s + (−0.119 − 0.992i)16-s − 0.747i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.892i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.452 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.452 + 0.892i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.452 + 0.892i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0758137 - 0.123412i\)
\(L(\frac12)\) \(\approx\) \(0.0758137 - 0.123412i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.15 - 2.32i)T \)
5 \( 1 + (-46.4 + 31.0i)T \)
good3 \( 1 + (9.83 - 9.83i)T - 243iT^{2} \)
7 \( 1 + 45.7T + 1.68e4T^{2} \)
11 \( 1 + (266. - 266. i)T - 1.61e5iT^{2} \)
13 \( 1 + (19.5 - 19.5i)T - 3.71e5iT^{2} \)
17 \( 1 + 891. iT - 1.41e6T^{2} \)
19 \( 1 + (334. + 334. i)T + 2.47e6iT^{2} \)
23 \( 1 + 1.57e3T + 6.43e6T^{2} \)
29 \( 1 + (116. + 116. i)T + 2.05e7iT^{2} \)
31 \( 1 + 3.23e3T + 2.86e7T^{2} \)
37 \( 1 + (3.95e3 + 3.95e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 9.81e3iT - 1.15e8T^{2} \)
43 \( 1 + (1.05e4 + 1.05e4i)T + 1.47e8iT^{2} \)
47 \( 1 + 3.83e3iT - 2.29e8T^{2} \)
53 \( 1 + (9.37e3 + 9.37e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (56.0 - 56.0i)T - 7.14e8iT^{2} \)
61 \( 1 + (-2.07e4 - 2.07e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (-5.44e3 + 5.44e3i)T - 1.35e9iT^{2} \)
71 \( 1 - 8.08e4iT - 1.80e9T^{2} \)
73 \( 1 + 7.17e4T + 2.07e9T^{2} \)
79 \( 1 + 9.38e4T + 3.07e9T^{2} \)
83 \( 1 + (-5.93e4 + 5.93e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.31e5iT - 5.58e9T^{2} \)
97 \( 1 - 7.43e4iT - 8.58e9T^{2} \)
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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

−13.06067557047262678716843043750, −11.66130911390871485988839113944, −10.37471583597085936160473286581, −9.860618655770756559369848852847, −8.707207249610446330242692493683, −7.23265149203504007395133644361, −5.79428905314704003633306980537, −4.91023599995591089802178187326, −2.10637620001883227457851236664, −0.085340120211804335407414893196, 1.59708713121174325163573065499, 3.19049930236186075801991337767, 5.92793323665471896983018344686, 6.71343451863836863203909886771, 8.075601012761156544374046768977, 9.484420693080892986886707484537, 10.47652693247964725024513394724, 11.38107207419405043654526178704, 12.57899090213628039782277075946, 13.34591742621980714595182056575

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