Properties

Degree $2$
Conductor $80$
Sign $-0.995 - 0.0963i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.839 − 0.839i)3-s + (3.71 − 55.7i)5-s + (−99.3 − 99.3i)7-s + 241. i·9-s + 637. i·11-s + (−640. − 640. i)13-s + (−43.7 − 49.9i)15-s + (648. − 648. i)17-s − 2.50e3·19-s − 166.·21-s + (−2.80e3 + 2.80e3i)23-s + (−3.09e3 − 414. i)25-s + (406. + 406. i)27-s − 4.95e3i·29-s + 1.96e3i·31-s + ⋯
L(s)  = 1  + (0.0538 − 0.0538i)3-s + (0.0664 − 0.997i)5-s + (−0.766 − 0.766i)7-s + 0.994i·9-s + 1.58i·11-s + (−1.05 − 1.05i)13-s + (−0.0501 − 0.0573i)15-s + (0.544 − 0.544i)17-s − 1.59·19-s − 0.0825·21-s + (−1.10 + 1.10i)23-s + (−0.991 − 0.132i)25-s + (0.107 + 0.107i)27-s − 1.09i·29-s + 0.366i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.995 - 0.0963i$
Motivic weight: \(5\)
Character: $\chi_{80} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.995 - 0.0963i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0126866 + 0.262718i\)
\(L(\frac12)\) \(\approx\) \(0.0126866 + 0.262718i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3.71 + 55.7i)T \)
good3 \( 1 + (-0.839 + 0.839i)T - 243iT^{2} \)
7 \( 1 + (99.3 + 99.3i)T + 1.68e4iT^{2} \)
11 \( 1 - 637. iT - 1.61e5T^{2} \)
13 \( 1 + (640. + 640. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-648. + 648. i)T - 1.41e6iT^{2} \)
19 \( 1 + 2.50e3T + 2.47e6T^{2} \)
23 \( 1 + (2.80e3 - 2.80e3i)T - 6.43e6iT^{2} \)
29 \( 1 + 4.95e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.96e3iT - 2.86e7T^{2} \)
37 \( 1 + (1.89e3 - 1.89e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 5.82e3T + 1.15e8T^{2} \)
43 \( 1 + (-1.06e4 + 1.06e4i)T - 1.47e8iT^{2} \)
47 \( 1 + (8.30e3 + 8.30e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (-7.43e3 - 7.43e3i)T + 4.18e8iT^{2} \)
59 \( 1 - 1.67e4T + 7.14e8T^{2} \)
61 \( 1 - 2.37e4T + 8.44e8T^{2} \)
67 \( 1 + (4.15e3 + 4.15e3i)T + 1.35e9iT^{2} \)
71 \( 1 + 1.22e4iT - 1.80e9T^{2} \)
73 \( 1 + (-3.60e4 - 3.60e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 6.43e4T + 3.07e9T^{2} \)
83 \( 1 + (-6.28e4 + 6.28e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 2.44e4iT - 5.58e9T^{2} \)
97 \( 1 + (8.96e4 - 8.96e4i)T - 8.58e9iT^{2} \)
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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

−12.86440317463055806576115075942, −12.09179902208466950747015736442, −10.24342422048495668512224257600, −9.768888899926958429085861383196, −8.088289453661232120047277641295, −7.18318502173427707999993614278, −5.35949865974750092615512394716, −4.23103460245013066499069696131, −2.10215449842015770000186826855, −0.10223058955846150327380687089, 2.53306642534811098433529980698, 3.78309609853274643158912870320, 6.04367377506683502529429924457, 6.63382242044878872268587214514, 8.442424276902036986466932128064, 9.514509788330840945036052623760, 10.64792459505544757220350788199, 11.84141379520036653035254905496, 12.75189687183818733602584457372, 14.30582191580298017031461664274

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