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Properties

Label 1-360-360.229-r0-0-0
Degree $1$
Conductor $360$
Sign $0.173 - 0.984i$
Analytic cond. $1.67183$
Root an. cond. $1.67183$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(1.67183\)
Root analytic conductor: \(1.67183\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 360,\ (0:\ ),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9002775232 - 0.7554225376i\)
\(L(\frac12)\) \(\approx\) \(0.9002775232 - 0.7554225376i\)
\(L(1)\) \(\approx\) \(1.006060147 - 0.2878600284i\)
\(L(1)\) \(\approx\) \(1.006060147 - 0.2878600284i\)

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

−24.99628257830276506549759601326, −24.08778128765792794309169419988, −23.252585724467076495875060315182, −22.06089143431394537267640013200, −21.634490454732091269818721161374, −20.51532380665483247228273881531, −19.65287396014983756462912995232, −18.70868052965138012344785115097, −17.84615676706844234266434804765, −17.03186421308155820774235461129, −15.985185930854309627976356023402, −14.84377993202715282054500008482, −14.52007943595540180281628547599, −13.061878925732818595767175372290, −12.226707588658441181847166002533, −11.41839595311933905271297790282, −10.345358821364340700424419586680, −9.1048656466369051911277129519, −8.58275000181833674494873966863, −7.13366401193149362449820802466, −6.37389538489927942079168150780, −4.957007189845355809181767793687, −4.27533832750375921331266776472, −2.58405025894135558986817207565, −1.71892802156255199432526524415, 0.72637560942667920345027054597, 2.23182210210250051054568765395, 3.62132409313093760570187282819, 4.55937231397428471695138160319, 5.772343949024741974145058699, 6.8712604347747582678484899199, 7.88382793505782605682539244499, 8.77954774519027641627626131536, 9.99063531536579532077030686156, 10.93551769302472484015549175649, 11.62749509432775809348172898993, 13.02045854303292889284195408951, 13.63537072707944674928840387858, 14.71989988193526054960958449097, 15.48063633389914908954993923769, 16.81522772011457833251339181143, 17.25570219058304712031359517562, 18.26545657508456178450496310041, 19.480440765544732118840346619868, 19.98831705236260646000725077372, 21.073328079449005816277934232166, 21.8699160857013024934639940922, 22.83027169359444374250484226521, 23.721992099227428332183196522800, 24.48815419833184892852913278030

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