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Properties

Label 1-360-360.347-r1-0-0
Degree $1$
Conductor $360$
Sign $0.665 - 0.746i$
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  + (0.866 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.665 - 0.746i$
Analytic conductor: \(38.6873\)
Root analytic conductor: \(38.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 360,\ (1:\ ),\ 0.665 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.005461243 - 0.8988199966i\)
\(L(\frac12)\) \(\approx\) \(2.005461243 - 0.8988199966i\)
\(L(1)\) \(\approx\) \(1.249541978 - 0.1575153233i\)
\(L(1)\) \(\approx\) \(1.249541978 - 0.1575153233i\)

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

−24.4578234017046411082136616646, −23.78083854548770805458591873171, −23.016571452111305148962112138, −21.94972578381086731918104047231, −21.02825917640192482613690827535, −20.343934211156128606081093476988, −19.392295334733882907237648186081, −18.37172295321961036351132688325, −17.44475929580150430098265157708, −16.83776870557967442649199510938, −15.61963121105902803299870160368, −14.68250351991261440290456422824, −13.9938630973916510214353600467, −12.865959724765815828426470308352, −11.92046197510650722509715602704, −10.90976741131701526204222863761, −10.15299335280143921468211687046, −8.83357471888603442941782349891, −8.06492842309581432345043922449, −6.90366199973229537159047454277, −5.981543450450811918354270730, −4.48821991590570158175957880672, −3.964668293964421319997766592363, −2.1690434548058438950941733597, −1.203034827003406919096495032, 0.69745550630985281868392863674, 2.01562822344604228618905689915, 3.30607903360754876633163887604, 4.50293143177496622566867653910, 5.6372820403408240023299719331, 6.49199233869125896444722542341, 7.96973732826119052114185814854, 8.54965916329224390352741283903, 9.64668807333800867149473528328, 10.96583869014573561805953518170, 11.50943281736548409422291537933, 12.58713968334018350729852794735, 13.763150647026865085353082165119, 14.40581558472898974127462405881, 15.546265282100801519694832522359, 16.24583252337177411801441653524, 17.48149881383859544003494911480, 18.11400625140175550273436121554, 19.06807897847087480916270591724, 19.98357516918791463615465547555, 21.13423734410602071068191794173, 21.513465115660011089301533950834, 22.72106641844403464529975539380, 23.50059383631894375145150741110, 24.586603911374240269995534833335

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