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Properties

Label 1-180-180.119-r0-0-0
Degree $1$
Conductor $180$
Sign $0.342 - 0.939i$
Analytic cond. $0.835916$
Root an. cond. $0.835916$
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

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

Invariants

Degree: \(1\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(0.835916\)
Root analytic conductor: \(0.835916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 180,\ (0:\ ),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8132614870 - 0.5694518237i\)
\(L(\frac12)\) \(\approx\) \(0.8132614870 - 0.5694518237i\)
\(L(1)\) \(\approx\) \(0.9424553606 - 0.2538749687i\)
\(L(1)\) \(\approx\) \(0.9424553606 - 0.2538749687i\)

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

−27.766383256644292535606389490524, −26.36211823466339694260977952108, −25.58668460334434362380163045889, −24.8637834131174703725262035490, −23.42775401614163894365262618434, −22.96196340905600902865996163234, −21.48633939153869672568560378632, −21.08473063415780744321646034264, −19.58440274168784408903561410462, −18.8543394459105857093044587474, −17.87818650461253480004234397673, −16.71277728735171898631308582918, −15.67400720792844415606271717081, −14.88136363030519084360387488709, −13.59652484468391765899678529630, −12.53222413683686250413244950762, −11.714232568686023063881998368140, −10.321887412739842635423742594312, −9.3511215780445861466263001036, −8.28040763929447618226531573002, −6.94143051731267387444982011292, −5.856253164341548960031422553456, −4.615662313709134543824626766799, −3.14838030034735028474594373299, −1.831181252524657654014537633443, 0.83103712433728346014967547012, 2.856531980769036553486637640554, 3.91624399203579866426284342576, 5.42257862547026493068073957241, 6.54100600714355538211985360064, 7.78160248134562047397367211174, 8.78593340831895684769143134763, 10.32846670263645634680939972273, 10.77414498878964203602280130083, 12.36972748029010097145931325595, 13.27159636850692968269197240088, 14.18198782953574104772481163911, 15.44552114948533291377363490337, 16.44988372313307082196905029683, 17.22938514442586849831630391666, 18.54816106508837633008257744178, 19.32268016907647316446690806206, 20.48423734811887234244125458373, 21.21451049933657134432803381491, 22.52245957282948230917842289638, 23.281838030661127205023922601475, 24.1519119510735641384082454425, 25.402687747058757906222202892999, 26.13313730506446909555509348668, 27.13593722502614934573153754737

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