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Properties

Label 2-80-80.67-c1-0-3
Degree $2$
Conductor $80$
Sign $0.980 - 0.196i$
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  + (−1.14 + 0.828i)2-s − 0.692i·3-s + (0.627 − 1.89i)4-s + (2.22 + 0.245i)5-s + (0.573 + 0.794i)6-s + (−0.343 − 0.343i)7-s + (0.853 + 2.69i)8-s + 2.52·9-s + (−2.75 + 1.55i)10-s + (0.843 + 0.843i)11-s + (−1.31 − 0.434i)12-s − 3.68·13-s + (0.678 + 0.109i)14-s + (0.169 − 1.53i)15-s + (−3.21 − 2.38i)16-s + (0.412 + 0.412i)17-s + ⋯
L(s)  = 1  + (−0.810 + 0.585i)2-s − 0.399i·3-s + (0.313 − 0.949i)4-s + (0.993 + 0.109i)5-s + (0.234 + 0.324i)6-s + (−0.129 − 0.129i)7-s + (0.301 + 0.953i)8-s + 0.840·9-s + (−0.869 + 0.493i)10-s + (0.254 + 0.254i)11-s + (−0.379 − 0.125i)12-s − 1.02·13-s + (0.181 + 0.0292i)14-s + (0.0438 − 0.397i)15-s + (−0.802 − 0.596i)16-s + (0.0999 + 0.0999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.980 - 0.196i$
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: \((2,\ 80,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.760534 + 0.0753991i\)
\(L(\frac12)\) \(\approx\) \(0.760534 + 0.0753991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 - 0.828i)T \)
5 \( 1 + (-2.22 - 0.245i)T \)
good3 \( 1 + 0.692iT - 3T^{2} \)
7 \( 1 + (0.343 + 0.343i)T + 7iT^{2} \)
11 \( 1 + (-0.843 - 0.843i)T + 11iT^{2} \)
13 \( 1 + 3.68T + 13T^{2} \)
17 \( 1 + (-0.412 - 0.412i)T + 17iT^{2} \)
19 \( 1 + (5.37 + 5.37i)T + 19iT^{2} \)
23 \( 1 + (3.08 - 3.08i)T - 23iT^{2} \)
29 \( 1 + (4.22 - 4.22i)T - 29iT^{2} \)
31 \( 1 - 8.75iT - 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
41 \( 1 - 2.54iT - 41T^{2} \)
43 \( 1 - 4.30T + 43T^{2} \)
47 \( 1 + (-4.56 + 4.56i)T - 47iT^{2} \)
53 \( 1 + 6.07iT - 53T^{2} \)
59 \( 1 + (7.33 - 7.33i)T - 59iT^{2} \)
61 \( 1 + (4.81 + 4.81i)T + 61iT^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 + 2.97T + 71T^{2} \)
73 \( 1 + (6.87 + 6.87i)T + 73iT^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 7.15iT - 83T^{2} \)
89 \( 1 - 1.10T + 89T^{2} \)
97 \( 1 + (-7.15 - 7.15i)T + 97iT^{2} \)
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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

−14.55035488780389464936975267720, −13.51069269286602401990485145169, −12.39084965769282036065335833702, −10.71243111293217889428016139248, −9.862236195444915600025779205066, −8.887365211808021751229348529468, −7.28048831174309041384774241786, −6.57390776362773085668724857712, −5.04345819430678989282960683638, −1.94286648681342546356333279966, 2.14103448986340814153092380725, 4.19306731332274611733261548386, 6.17392987682462653714684503891, 7.68575851574117954396187097044, 9.142674657050518975885936962878, 9.920036285460541616597579202672, 10.67484925908879209136033621177, 12.22941768190475132503091981162, 12.96396490285967532589055366289, 14.32911196938981672369923699334

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