Properties

Label 2-80-16.5-c5-0-10
Degree $2$
Conductor $80$
Sign $-0.325 - 0.945i$
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  + (−2.24 + 5.19i)2-s + (1.65 − 1.65i)3-s + (−21.9 − 23.3i)4-s + (−17.6 − 17.6i)5-s + (4.87 + 12.3i)6-s − 111. i·7-s + (170. − 61.5i)8-s + 237. i·9-s + (131. − 52.1i)10-s + (263. + 263. i)11-s + (−74.8 − 2.28i)12-s + (−480. + 480. i)13-s + (577. + 249. i)14-s − 58.5·15-s + (−62.5 + 1.02e3i)16-s + 1.41e3·17-s + ⋯
L(s)  = 1  + (−0.396 + 0.917i)2-s + (0.106 − 0.106i)3-s + (−0.685 − 0.728i)4-s + (−0.316 − 0.316i)5-s + (0.0553 + 0.139i)6-s − 0.858i·7-s + (0.940 − 0.339i)8-s + 0.977i·9-s + (0.415 − 0.164i)10-s + (0.656 + 0.656i)11-s + (−0.150 − 0.00458i)12-s + (−0.788 + 0.788i)13-s + (0.787 + 0.340i)14-s − 0.0671·15-s + (−0.0611 + 0.998i)16-s + 1.18·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.325 - 0.945i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.325 - 0.945i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.657702 + 0.922013i\)
\(L(\frac12)\) \(\approx\) \(0.657702 + 0.922013i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.24 - 5.19i)T \)
5 \( 1 + (17.6 + 17.6i)T \)
good3 \( 1 + (-1.65 + 1.65i)T - 243iT^{2} \)
7 \( 1 + 111. iT - 1.68e4T^{2} \)
11 \( 1 + (-263. - 263. i)T + 1.61e5iT^{2} \)
13 \( 1 + (480. - 480. i)T - 3.71e5iT^{2} \)
17 \( 1 - 1.41e3T + 1.41e6T^{2} \)
19 \( 1 + (-555. + 555. i)T - 2.47e6iT^{2} \)
23 \( 1 - 3.84e3iT - 6.43e6T^{2} \)
29 \( 1 + (2.77e3 - 2.77e3i)T - 2.05e7iT^{2} \)
31 \( 1 + 5.16e3T + 2.86e7T^{2} \)
37 \( 1 + (-7.57e3 - 7.57e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 3.02e3iT - 1.15e8T^{2} \)
43 \( 1 + (-1.01e4 - 1.01e4i)T + 1.47e8iT^{2} \)
47 \( 1 - 2.51e4T + 2.29e8T^{2} \)
53 \( 1 + (2.13e4 + 2.13e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-2.48e4 - 2.48e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-5.60e3 + 5.60e3i)T - 8.44e8iT^{2} \)
67 \( 1 + (1.91e4 - 1.91e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 1.79e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.45e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.63e4T + 3.07e9T^{2} \)
83 \( 1 + (5.92e3 - 5.92e3i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.24e5iT - 5.58e9T^{2} \)
97 \( 1 - 5.05e4T + 8.58e9T^{2} \)
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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

−14.01058003410887884912927774258, −12.95173661971688627975433351269, −11.46192215052188266974593091908, −10.08403844629579758224546503767, −9.179269877028639878575562971362, −7.61621255923831561664509534213, −7.21921929718893052045542720483, −5.36255417228874053090628891863, −4.16938667152077540488081167936, −1.36744441868663967174533068117, 0.61884746589203690598406408794, 2.69286101504178109564569163705, 3.85711600801432447570253553333, 5.76115682546991128359271302050, 7.59550680784504416062657320521, 8.819613819324938595934329138909, 9.706237978669797769598529036254, 10.90855155037242216781897561722, 12.13599278117419461479645331124, 12.48468652045402864868107633030

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