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Properties

Label 2-80-80.27-c1-0-1
Degree $2$
Conductor $80$
Sign $0.971 - 0.237i$
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  + (−1.23 + 0.687i)2-s + 0.614·3-s + (1.05 − 1.69i)4-s + (0.832 − 2.07i)5-s + (−0.759 + 0.422i)6-s + (2.83 + 2.83i)7-s + (−0.134 + 2.82i)8-s − 2.62·9-s + (0.399 + 3.13i)10-s + (1.95 − 1.95i)11-s + (0.647 − 1.04i)12-s + 2.05i·13-s + (−5.45 − 1.55i)14-s + (0.511 − 1.27i)15-s + (−1.77 − 3.58i)16-s + (−4.06 − 4.06i)17-s + ⋯
L(s)  = 1  + (−0.873 + 0.486i)2-s + 0.354·3-s + (0.527 − 0.849i)4-s + (0.372 − 0.928i)5-s + (−0.310 + 0.172i)6-s + (1.07 + 1.07i)7-s + (−0.0473 + 0.998i)8-s − 0.874·9-s + (0.126 + 0.992i)10-s + (0.590 − 0.590i)11-s + (0.187 − 0.301i)12-s + 0.569i·13-s + (−1.45 − 0.415i)14-s + (0.132 − 0.329i)15-s + (−0.444 − 0.895i)16-s + (−0.986 − 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.971 - 0.237i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.971 - 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781323 + 0.0942148i\)
\(L(\frac12)\) \(\approx\) \(0.781323 + 0.0942148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 - 0.687i)T \)
5 \( 1 + (-0.832 + 2.07i)T \)
good3 \( 1 - 0.614T + 3T^{2} \)
7 \( 1 + (-2.83 - 2.83i)T + 7iT^{2} \)
11 \( 1 + (-1.95 + 1.95i)T - 11iT^{2} \)
13 \( 1 - 2.05iT - 13T^{2} \)
17 \( 1 + (4.06 + 4.06i)T + 17iT^{2} \)
19 \( 1 + (0.683 - 0.683i)T - 19iT^{2} \)
23 \( 1 + (4.95 - 4.95i)T - 23iT^{2} \)
29 \( 1 + (-0.835 - 0.835i)T + 29iT^{2} \)
31 \( 1 - 2.35iT - 31T^{2} \)
37 \( 1 + 4.54iT - 37T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 - 0.849iT - 43T^{2} \)
47 \( 1 + (2.72 - 2.72i)T - 47iT^{2} \)
53 \( 1 - 5.17T + 53T^{2} \)
59 \( 1 + (4.16 + 4.16i)T + 59iT^{2} \)
61 \( 1 + (-5.55 + 5.55i)T - 61iT^{2} \)
67 \( 1 + 1.73iT - 67T^{2} \)
71 \( 1 - 2.33T + 71T^{2} \)
73 \( 1 + (4.39 + 4.39i)T + 73iT^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + 2.75T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (3.52 + 3.52i)T + 97iT^{2} \)
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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

−14.46541710734240987943050419243, −13.80048501264118986621730039604, −11.87600653183774823068494929471, −11.29006862335881686896910388042, −9.375284214136735821386370507987, −8.830481126399657237327724660157, −8.007839691380911342159656289646, −6.12534650843200711175794983341, −5.06200471214124566186870574615, −2.02478270530080303212050181255, 2.18868315543977906180666127311, 3.97683909933644992270081367558, 6.48263173900736652104586513953, 7.70333712897086316375362303218, 8.665660243383505167411829667715, 10.16997819110150421036859618475, 10.82827710763826552045010407533, 11.79039006071561286384516068401, 13.36371034760619877991075142762, 14.40428974531642593672888197354

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