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Properties

Label 2-80-80.67-c1-0-0
Degree $2$
Conductor $80$
Sign $-0.637 - 0.770i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 + 0.430i)2-s + 2.96i·3-s + (1.62 − 1.15i)4-s + (−2.22 + 0.177i)5-s + (−1.27 − 3.99i)6-s + (−0.115 − 0.115i)7-s + (−1.69 + 2.26i)8-s − 5.79·9-s + (2.92 − 1.19i)10-s + (2.95 + 2.95i)11-s + (3.43 + 4.83i)12-s + 1.55·13-s + (0.204 + 0.105i)14-s + (−0.525 − 6.61i)15-s + (1.31 − 3.77i)16-s + (0.299 + 0.299i)17-s + ⋯
L(s)  = 1  + (−0.952 + 0.304i)2-s + 1.71i·3-s + (0.814 − 0.579i)4-s + (−0.996 + 0.0793i)5-s + (−0.520 − 1.63i)6-s + (−0.0435 − 0.0435i)7-s + (−0.599 + 0.800i)8-s − 1.93·9-s + (0.925 − 0.378i)10-s + (0.892 + 0.892i)11-s + (0.992 + 1.39i)12-s + 0.432·13-s + (0.0546 + 0.0282i)14-s + (−0.135 − 1.70i)15-s + (0.327 − 0.944i)16-s + (0.0726 + 0.0726i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.637 - 0.770i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ -0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.235694 + 0.501155i\)
\(L(\frac12)\) \(\approx\) \(0.235694 + 0.501155i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.430i)T \)
5 \( 1 + (2.22 - 0.177i)T \)
good3 \( 1 - 2.96iT - 3T^{2} \)
7 \( 1 + (0.115 + 0.115i)T + 7iT^{2} \)
11 \( 1 + (-2.95 - 2.95i)T + 11iT^{2} \)
13 \( 1 - 1.55T + 13T^{2} \)
17 \( 1 + (-0.299 - 0.299i)T + 17iT^{2} \)
19 \( 1 + (-2.26 - 2.26i)T + 19iT^{2} \)
23 \( 1 + (-4.14 + 4.14i)T - 23iT^{2} \)
29 \( 1 + (0.289 - 0.289i)T - 29iT^{2} \)
31 \( 1 - 4.18iT - 31T^{2} \)
37 \( 1 - 1.63T + 37T^{2} \)
41 \( 1 + 7.61iT - 41T^{2} \)
43 \( 1 + 6.72T + 43T^{2} \)
47 \( 1 + (4.38 - 4.38i)T - 47iT^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + (1.63 - 1.63i)T - 59iT^{2} \)
61 \( 1 + (1.23 + 1.23i)T + 61iT^{2} \)
67 \( 1 - 2.49T + 67T^{2} \)
71 \( 1 - 8.00T + 71T^{2} \)
73 \( 1 + (-1.12 - 1.12i)T + 73iT^{2} \)
79 \( 1 + 3.62T + 79T^{2} \)
83 \( 1 + 1.62iT - 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + (-9.69 - 9.69i)T + 97iT^{2} \)
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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

−14.99365918692311576816285144057, −14.54888207667535586665276584678, −12.07402864300686661499991046723, −11.12506013150074619072091714100, −10.24018401830656735557880392071, −9.260697758712646284264909844080, −8.324141314784163581911549430843, −6.79520217848282848916893588895, −4.97531332359176309055620556335, −3.56168342163255562777626767352, 1.07650061040305668188564730603, 3.20186047640124971935993350644, 6.28337041235252234115941030529, 7.29789643498546569957732131233, 8.159254017763511876507788307744, 9.124581514170267242516462900865, 11.28303940405842313284548993934, 11.60282036406157116285470985844, 12.69088125615007132745178510924, 13.67457800793446000096858793390

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