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Properties

Label 1-360-360.139-r1-0-0
Degree $1$
Conductor $360$
Sign $0.173 - 0.984i$
Analytic cond. $38.6873$
Root an. cond. $38.6873$
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

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

Invariants

Degree: \(1\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(38.6873\)
Root analytic conductor: \(38.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 360,\ (1:\ ),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7545647904 - 0.6331550373i\)
\(L(\frac12)\) \(\approx\) \(0.7545647904 - 0.6331550373i\)
\(L(1)\) \(\approx\) \(0.8760418234 + 0.01366094535i\)
\(L(1)\) \(\approx\) \(0.8760418234 + 0.01366094535i\)

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

−24.47738989862722660275778909627, −23.92323896989489465628687382233, −23.0106073267673394127446692846, −22.016122598846141978634552844011, −21.32967936600059558802233057395, −20.09689023115988320095292717578, −19.59980646807377634394791853297, −18.52693425650285946334597597003, −17.62102480040279869092938969285, −16.50376191141954490516136768375, −16.063330840382913958407163091843, −14.813544762608480751033265802036, −13.64637269799699213505426776900, −13.32827772437746075046094811900, −11.89457230810934355094381980345, −11.0892105494119488568855490982, −10.03342761261333729129832535929, −9.20029007957765756599183717603, −7.9634805307071029262832164640, −7.03304571963023280957450503480, −6.06914874806236214095080326289, −4.78911915717114084474258333855, −3.73465502336218556615049875004, −2.60715030552029290671895051921, −1.03894187482345115806990544621, 0.31406775801610124732568812253, 2.16735642181852197137026420672, 2.98032664827181222729264539366, 4.50393794186864847163511775277, 5.463635125289997431239372502167, 6.53022255255249710051683486858, 7.621128083095005243336312540598, 8.652553358674886898799682835489, 9.706322105675302124821419544419, 10.44415667191776105205508226839, 11.8037402337940970626689959315, 12.51763656767191364538060386528, 13.35342114381683188203861847733, 14.578926227246127249945011639209, 15.5311026777503690416181075502, 15.9916269985584325528082612155, 17.45969996327255262704469916920, 18.02557943451679864762605647259, 18.99800165297557403767313788879, 20.006702030549534238083423682908, 20.6595036583223531744014302131, 21.94480835607491226288578811674, 22.43379837183736955974655052752, 23.32706424928651609933001450991, 24.586710670102420903489642804621

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