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Properties

Label 2-80-80.67-c1-0-2
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: $\chi_{80} (67, \cdot )$
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

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

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