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Properties

Label 2-80-80.3-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: $\chi_{80} (3, \cdot )$
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

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

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