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Properties

Label 1-180-180.79-r1-0-0
Degree $1$
Conductor $180$
Sign $-0.939 - 0.342i$
Analytic cond. $19.3436$
Root an. cond. $19.3436$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s − 17-s − 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s − 37-s + (−0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s − 53-s + (0.5 − 0.866i)59-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s − 17-s − 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s − 37-s + (−0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s − 53-s + (0.5 − 0.866i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(19.3436\)
Root analytic conductor: \(19.3436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 180,\ (1:\ ),\ -0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08478323559 - 0.4808296226i\)
\(L(\frac12)\) \(\approx\) \(0.08478323559 - 0.4808296226i\)
\(L(1)\) \(\approx\) \(0.7988324569 - 0.1408557152i\)
\(L(1)\) \(\approx\) \(0.7988324569 - 0.1408557152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.57859346429966183339175390428, −26.4645383218042554060981632316, −25.65227179001161090697899407043, −24.60058636052511889580135010143, −23.844358075256018180414021159964, −22.531269835120791810700985771141, −21.82403316264162670571406837516, −20.92218415878567481242816255630, −19.54617123816639698301163655354, −18.914207348125614932957405010465, −17.8878678587834677466910754327, −16.59109130592089197806180462087, −15.87396599901817723316239051464, −14.71600711785655947296565670013, −13.6721025743400107090446660419, −12.57559554895601309269002250059, −11.568438768182544050799621497170, −10.531050582491404991413013743, −9.00884084848137959601367515413, −8.58867019905235862496014227127, −6.72544624423465379238770853777, −6.04386621655767266651217941899, −4.520566899505271874868067696847, −3.205884482504983972106960170806, −1.81263338074527618888872419919, 0.16397356514340721383071475628, 1.86310405224292028472763200672, 3.53370181755788688118107496246, 4.53053707662123979157856388912, 6.122493576298045876595819528, 7.088258551572188526337351203, 8.26725489011218214527040361026, 9.60708230976048022713222515821, 10.468048540135157515659885054250, 11.61641561981770368198516276707, 12.93551838266130740775020989541, 13.58055051266230690330255996100, 14.96068047031785287236709961494, 15.766876553576468120000113133060, 17.09388332101134560705811282642, 17.64784832315501990781673800415, 19.0342556856564974758541191467, 20.0053993883633338216994786922, 20.63269742018073258746592229234, 22.03780853303737108437670036865, 22.870811428529201163897847700083, 23.62860733978518762047928990377, 24.8747659097786287075334677222, 25.742198680402967679227680981559, 26.576041018024390109539947702327

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