Properties

Label 2-80-80.43-c1-0-4
Degree $2$
Conductor $80$
Sign $0.987 + 0.158i$
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  + (−0.558 + 1.29i)2-s − 2.55i·3-s + (−1.37 − 1.45i)4-s + (1.66 + 1.49i)5-s + (3.31 + 1.42i)6-s + (2.40 − 2.40i)7-s + (2.65 − 0.977i)8-s − 3.51·9-s + (−2.86 + 1.33i)10-s + (−2.67 + 2.67i)11-s + (−3.70 + 3.51i)12-s − 2.40·13-s + (1.78 + 4.46i)14-s + (3.80 − 4.25i)15-s + (−0.212 + 3.99i)16-s + (−0.0750 + 0.0750i)17-s + ⋯
L(s)  = 1  + (−0.394 + 0.918i)2-s − 1.47i·3-s + (−0.688 − 0.725i)4-s + (0.745 + 0.666i)5-s + (1.35 + 0.581i)6-s + (0.908 − 0.908i)7-s + (0.938 − 0.345i)8-s − 1.17·9-s + (−0.906 + 0.421i)10-s + (−0.807 + 0.807i)11-s + (−1.06 + 1.01i)12-s − 0.666·13-s + (0.475 + 1.19i)14-s + (0.982 − 1.09i)15-s + (−0.0532 + 0.998i)16-s + (−0.0182 + 0.0182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.987 + 0.158i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.987 + 0.158i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841292 - 0.0668840i\)
\(L(\frac12)\) \(\approx\) \(0.841292 - 0.0668840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.558 - 1.29i)T \)
5 \( 1 + (-1.66 - 1.49i)T \)
good3 \( 1 + 2.55iT - 3T^{2} \)
7 \( 1 + (-2.40 + 2.40i)T - 7iT^{2} \)
11 \( 1 + (2.67 - 2.67i)T - 11iT^{2} \)
13 \( 1 + 2.40T + 13T^{2} \)
17 \( 1 + (0.0750 - 0.0750i)T - 17iT^{2} \)
19 \( 1 + (2.67 - 2.67i)T - 19iT^{2} \)
23 \( 1 + (-2.12 - 2.12i)T + 23iT^{2} \)
29 \( 1 + (-3.95 - 3.95i)T + 29iT^{2} \)
31 \( 1 + 1.65iT - 31T^{2} \)
37 \( 1 + 2.53T + 37T^{2} \)
41 \( 1 - 1.70iT - 41T^{2} \)
43 \( 1 + 3.84T + 43T^{2} \)
47 \( 1 + (2.15 + 2.15i)T + 47iT^{2} \)
53 \( 1 + 1.29iT - 53T^{2} \)
59 \( 1 + (5.29 + 5.29i)T + 59iT^{2} \)
61 \( 1 + (-10.2 + 10.2i)T - 61iT^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 2.27T + 71T^{2} \)
73 \( 1 + (-9.99 + 9.99i)T - 73iT^{2} \)
79 \( 1 - 8.70T + 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (-5.00 + 5.00i)T - 97iT^{2} \)
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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

−14.27538869463194237372465510496, −13.56818497001172337823794405411, −12.60680227207723974734069439915, −10.87753298269392305479380559975, −9.888760301136494647752672197200, −8.118202775510730653522302479061, −7.34334494846344985292265259130, −6.57802826279991862568746990341, −5.06529010916204654717178906723, −1.79651105004497366420481368959, 2.59908451284629552398595266078, 4.59240293109881902141096687701, 5.35120136630378144989323204920, 8.371454947325230992646846245412, 8.992987706537944972338345601477, 10.06868614110452266456789453191, 10.88721153502412561291735931602, 11.98858946229418076420931128694, 13.22150039859252278212194070952, 14.42300738798830419907214425520

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