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Properties

Label 1-180-180.47-r1-0-0
Degree $1$
Conductor $180$
Sign $0.784 - 0.619i$
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  + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)13-s − i·17-s + 19-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + i·37-s + (0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + (0.866 − 0.5i)47-s + (0.5 − 0.866i)49-s + i·53-s + (0.5 − 0.866i)59-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)13-s − i·17-s + 19-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + i·37-s + (0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + (0.866 − 0.5i)47-s + (0.5 − 0.866i)49-s + i·53-s + (0.5 − 0.866i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.784 - 0.619i$
Analytic conductor: \(19.3436\)
Root analytic conductor: \(19.3436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 180,\ (1:\ ),\ 0.784 - 0.619i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.479869032 - 0.5136444519i\)
\(L(\frac12)\) \(\approx\) \(1.479869032 - 0.5136444519i\)
\(L(1)\) \(\approx\) \(1.043451759 - 0.08849371533i\)
\(L(1)\) \(\approx\) \(1.043451759 - 0.08849371533i\)

Euler product

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

−27.10908192340823863864551836900, −26.076072551773557933878611155990, −25.52345374357130021754457264786, −24.29916573755786169721182546691, −23.10213800790788247216681079428, −22.71220407816227164964728849802, −21.34315303687395339459030765193, −20.359176588607424748021449761646, −19.56938663829463714064455165708, −18.40276715486207764800325399611, −17.49283327295771658407216508106, −16.3196118212826222546225993957, −15.53851374695823155609188739747, −14.358871942628989878134161811004, −13.09558284834821595045332405751, −12.56255274697277429045072213750, −10.95835710101646162705262655400, −10.15559400135017449678490151934, −9.02413501426195955297964690766, −7.70140507178774807606618863989, −6.67526478490798260422654302743, −5.45491759007323316349016645225, −4.02101043289013070961191512963, −2.87109369781343863732487388850, −1.09370075853446726672101437647, 0.68940018828135221970862402498, 2.605920612263359156397060565024, 3.6572625302945163355149028573, 5.32022386972837441866033132452, 6.26542160051912758234654353932, 7.53023088181690996225356955069, 8.86683768436211835783215427324, 9.666832607640879158177106184017, 11.05515347861999760957025931614, 11.935922708549603310192569094927, 13.28429132197758417501783605572, 13.86613717531585449036210669143, 15.51233494915466185682661832386, 16.01781602284161114743417578412, 17.12827638805815676577969127773, 18.67564718527536671564488634139, 18.84167849860187113129316903478, 20.324339687289877086665576539341, 21.19799593466109648793547987680, 22.26238627848002923237485669582, 23.05472156721509928428999016022, 24.17747783238328506207497481546, 25.10566864493721491576328331025, 26.06797851560483109376390477873, 26.85211769077050595796131108159

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