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Properties

Label 2-80-20.3-c5-0-8
Degree $2$
Conductor $80$
Sign $0.930 + 0.365i$
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  + (−12.4 + 12.4i)3-s + (55 − 10i)5-s + (−136. − 136. i)7-s − 67i·9-s − 124. i·11-s + (165 + 165i)13-s + (−560. + 809. i)15-s + (1.26e3 − 1.26e3i)17-s + 2.73e3·19-s + 3.40e3·21-s + (535. − 535. i)23-s + (2.92e3 − 1.10e3i)25-s + (−2.19e3 − 2.19e3i)27-s − 2.59e3i·29-s + 7.09e3i·31-s + ⋯
L(s)  = 1  + (−0.798 + 0.798i)3-s + (0.983 − 0.178i)5-s + (−1.05 − 1.05i)7-s − 0.275i·9-s − 0.310i·11-s + (0.270 + 0.270i)13-s + (−0.642 + 0.928i)15-s + (1.06 − 1.06i)17-s + 1.74·19-s + 1.68·21-s + (0.211 − 0.211i)23-s + (0.936 − 0.352i)25-s + (−0.578 − 0.578i)27-s − 0.573i·29-s + 1.32i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.930 + 0.365i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 0.930 + 0.365i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.37902 - 0.260726i\)
\(L(\frac12)\) \(\approx\) \(1.37902 - 0.260726i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-55 + 10i)T \)
good3 \( 1 + (12.4 - 12.4i)T - 243iT^{2} \)
7 \( 1 + (136. + 136. i)T + 1.68e4iT^{2} \)
11 \( 1 + 124. iT - 1.61e5T^{2} \)
13 \( 1 + (-165 - 165i)T + 3.71e5iT^{2} \)
17 \( 1 + (-1.26e3 + 1.26e3i)T - 1.41e6iT^{2} \)
19 \( 1 - 2.73e3T + 2.47e6T^{2} \)
23 \( 1 + (-535. + 535. i)T - 6.43e6iT^{2} \)
29 \( 1 + 2.59e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.09e3iT - 2.86e7T^{2} \)
37 \( 1 + (-1.38e3 + 1.38e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 2.17e3T + 1.15e8T^{2} \)
43 \( 1 + (-1.38e4 + 1.38e4i)T - 1.47e8iT^{2} \)
47 \( 1 + (-4.94e3 - 4.94e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (2.39e4 + 2.39e4i)T + 4.18e8iT^{2} \)
59 \( 1 - 2.71e4T + 7.14e8T^{2} \)
61 \( 1 + 3.58e4T + 8.44e8T^{2} \)
67 \( 1 + (2.68e4 + 2.68e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 6.71e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.16e4 + 2.16e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 2.19e4T + 3.07e9T^{2} \)
83 \( 1 + (3.87e4 - 3.87e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 1.14e5iT - 5.58e9T^{2} \)
97 \( 1 + (615 - 615i)T - 8.58e9iT^{2} \)
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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

−13.58126268357849546634594184894, −12.19117839831421385593797163434, −10.88728649709343457794911331609, −9.999418050866135788733090715744, −9.377356327384543303621351341162, −7.30236450633331926870496724452, −5.98327888414176246780408086270, −4.94517258913932197617293002981, −3.30717702793803804894175151703, −0.78056945840489030741637433925, 1.26956502479806605577369791237, 3.00138161999955207109506749530, 5.69988475330536098507945134734, 6.02739870774811679023611141741, 7.39630311493647599852319939469, 9.197179710339818810567323010572, 10.04009963988037189877758011167, 11.53240625012714690339100853235, 12.56838551865365555237645020715, 13.07295323779353204209029441339

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