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Properties

Label 2-80-80.69-c1-0-4
Degree $2$
Conductor $80$
Sign $0.755 - 0.655i$
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 + i)2-s + (−1 − i)3-s + 2i·4-s + (2 + i)5-s − 2i·6-s + (−2 + 2i)8-s − i·9-s + (1 + 3i)10-s + (−3 − 3i)11-s + (2 − 2i)12-s + (−3 − 3i)13-s + (−1 − 3i)15-s − 4·16-s + 4i·17-s + (1 − i)18-s + (−1 + i)19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.577 − 0.577i)3-s + i·4-s + (0.894 + 0.447i)5-s − 0.816i·6-s + (−0.707 + 0.707i)8-s − 0.333i·9-s + (0.316 + 0.948i)10-s + (−0.904 − 0.904i)11-s + (0.577 − 0.577i)12-s + (−0.832 − 0.832i)13-s + (−0.258 − 0.774i)15-s − 16-s + 0.970i·17-s + (0.235 − 0.235i)18-s + (−0.229 + 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10789 + 0.413506i\)
\(L(\frac12)\) \(\approx\) \(1.10789 + 0.413506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 + (-2 - i)T \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + (3 + 3i)T + 11iT^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + (1 - i)T - 19iT^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (3 - 3i)T - 43iT^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + (-9 + 9i)T - 53iT^{2} \)
59 \( 1 + (-9 - 9i)T + 59iT^{2} \)
61 \( 1 + (5 - 5i)T - 61iT^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (9 + 9i)T + 83iT^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−14.57742198816264828588736620394, −13.21263327872961666426845699339, −12.85605969118348066594036764241, −11.49524970232935581612553876823, −10.26075897927273957609018588390, −8.550786391460375997190461702687, −7.18837648543051429883700073371, −6.13044312346254729524365106867, −5.28560249338780965286128836829, −3.00055124653600007170672131703, 2.35800388573391443252054775612, 4.81303490109649790741175582599, 5.16957075528633276263265883990, 6.91565115638612820206505696023, 9.228785443790332110863380008958, 10.07059388567639789516193372782, 10.96056544901509384935604857787, 12.17801621362332164398493210095, 13.12580103815786168627567314299, 14.04901381589848517669287105152

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