Properties

Label 2-80-80.67-c1-0-4
Degree $2$
Conductor $80$
Sign $0.168 - 0.985i$
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  + (0.759 + 1.19i)2-s + 1.39i·3-s + (−0.846 + 1.81i)4-s + (0.535 − 2.17i)5-s + (−1.66 + 1.05i)6-s + (−2.13 − 2.13i)7-s + (−2.80 + 0.366i)8-s + 1.05·9-s + (2.99 − 1.01i)10-s + (2.17 + 2.17i)11-s + (−2.52 − 1.17i)12-s + 1.54·13-s + (0.925 − 4.16i)14-s + (3.02 + 0.745i)15-s + (−2.56 − 3.06i)16-s + (−3.86 − 3.86i)17-s + ⋯
L(s)  = 1  + (0.536 + 0.843i)2-s + 0.804i·3-s + (−0.423 + 0.905i)4-s + (0.239 − 0.970i)5-s + (−0.678 + 0.431i)6-s + (−0.806 − 0.806i)7-s + (−0.991 + 0.129i)8-s + 0.353·9-s + (0.947 − 0.319i)10-s + (0.654 + 0.654i)11-s + (−0.728 − 0.340i)12-s + 0.428·13-s + (0.247 − 1.11i)14-s + (0.780 + 0.192i)15-s + (−0.641 − 0.766i)16-s + (−0.937 − 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.168 - 0.985i$
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: \((2,\ 80,\ (\ :1/2),\ 0.168 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.882974 + 0.744610i\)
\(L(\frac12)\) \(\approx\) \(0.882974 + 0.744610i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.759 - 1.19i)T \)
5 \( 1 + (-0.535 + 2.17i)T \)
good3 \( 1 - 1.39iT - 3T^{2} \)
7 \( 1 + (2.13 + 2.13i)T + 7iT^{2} \)
11 \( 1 + (-2.17 - 2.17i)T + 11iT^{2} \)
13 \( 1 - 1.54T + 13T^{2} \)
17 \( 1 + (3.86 + 3.86i)T + 17iT^{2} \)
19 \( 1 + (0.0136 + 0.0136i)T + 19iT^{2} \)
23 \( 1 + (3.15 - 3.15i)T - 23iT^{2} \)
29 \( 1 + (3.33 - 3.33i)T - 29iT^{2} \)
31 \( 1 + 8.92iT - 31T^{2} \)
37 \( 1 - 7.24T + 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + 2.02T + 43T^{2} \)
47 \( 1 + (3.34 - 3.34i)T - 47iT^{2} \)
53 \( 1 - 7.30iT - 53T^{2} \)
59 \( 1 + (-3.52 + 3.52i)T - 59iT^{2} \)
61 \( 1 + (-1.41 - 1.41i)T + 61iT^{2} \)
67 \( 1 - 0.748T + 67T^{2} \)
71 \( 1 + 0.269T + 71T^{2} \)
73 \( 1 + (0.811 + 0.811i)T + 73iT^{2} \)
79 \( 1 - 2.80T + 79T^{2} \)
83 \( 1 + 12.8iT - 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (-6.33 - 6.33i)T + 97iT^{2} \)
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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

−14.79630075931568612889052480062, −13.42517302196962976804928287427, −12.99069032891268509814796183583, −11.56854884245416954127529104293, −9.740714741269057855593157499125, −9.221806109058678684740571154958, −7.57205385146799323431187689089, −6.29630038788039366025039838134, −4.71880777950348626402217786536, −3.90143342478921247978563678283, 2.17416218862061596664810628327, 3.71059470283862852366981585869, 6.02548392547115128554610123314, 6.64615192626781552104619187530, 8.697024876674212192017550470070, 9.972568622287418811411631121407, 11.05188722666484513474238579127, 12.13903424145145382927399247252, 13.02967275712517779760318348648, 13.84714049698878547553928508879

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