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Properties

Label 2-80-20.3-c5-0-10
Degree $2$
Conductor $80$
Sign $0.414 + 0.910i$
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  + (−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

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.414 + 0.910i$
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: \((2,\ 80,\ (\ :5/2),\ 0.414 + 0.910i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.47277 - 0.947854i\)
\(L(\frac12)\) \(\approx\) \(1.47277 - 0.947854i\)
\(L(\frac{7}{2})\) 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

−13.25264407884723693602533888282, −12.06264645195827962780486502634, −11.19088812041694050316884384991, −9.785140862536597006803888580369, −8.489129526083448492973645712974, −7.79500555179221310814671903021, −5.52573842009209667744421165603, −5.01307465394410649171870395809, −2.75025390194078879097701405842, −0.814393730904499625618036560738, 1.62392508785158906151156088388, 3.53203457877552295268672087699, 5.03801979696096325484702216281, 7.03567319349742320860971262322, 7.31593780982687139751433393825, 9.395332757940644861794215012868, 10.20772834558806615139921801300, 11.51798082783768690236251966345, 12.29925243540734540904521146637, 13.89542710180201983757692756494

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