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Properties

Label 1-360-360.77-r0-0-0
Degree $1$
Conductor $360$
Sign $0.665 - 0.746i$
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  + (−0.866 − 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

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

Invariants

Degree: \(1\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.665 - 0.746i$
Analytic conductor: \(1.67183\)
Root analytic conductor: \(1.67183\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 360,\ (0:\ ),\ 0.665 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.026811137 - 0.4602025525i\)
\(L(\frac12)\) \(\approx\) \(1.026811137 - 0.4602025525i\)
\(L(1)\) \(\approx\) \(0.9863852189 - 0.1523229111i\)
\(L(1)\) \(\approx\) \(0.9863852189 - 0.1523229111i\)

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

−24.94684871638556316198912869144, −23.8370898257216203462591325520, −23.24010890907457878783070344277, −22.03428352293702323774051792407, −21.53594099415105145658254231307, −20.47796026314736172629590249710, −19.383776782750333678175339818705, −18.77650352229310801112379239403, −17.90838998924729907783548451785, −16.62066088109514159129248208058, −16.00923338474217714967987141370, −15.171781713972510208415100323643, −13.90565864072249435364526401091, −13.18768786034139006963867109253, −12.23638461383545976655664626573, −11.17377554072112105159558104525, −10.28508022897226987218230352139, −9.0946360285794972313213417231, −8.444760548148538779106177442195, −7.0791894481841528871339728875, −6.09424485487515160014119074864, −5.251615975479994817051389312532, −3.669965539167675598758802644510, −2.922893376240826048888661637, −1.31425351057890794825690200133, 0.78496462696082687053140724138, 2.51780859613820247743582177499, 3.53999035223922622063448633796, 4.74230053474827292543402988075, 5.88234317477948403452025458090, 7.0093051180403177343595374334, 7.77175413535522351458081201486, 9.153924974714155117998501328564, 9.93672363101772582289472497642, 10.85383073010542474370702496467, 11.972880607223023773003738446477, 13.06041356159441177222188467692, 13.578886888900382069143459316767, 14.83877346927266084030663296059, 15.834581424693261077805807708866, 16.41982367744613394283474094789, 17.636652769223621829366899074321, 18.364013144902134852528268479958, 19.34748263392904874393336517222, 20.431603012576078757492026695489, 20.7625975046039001698672965647, 22.25285249313135931172010094072, 22.90519023626783495635756693102, 23.47285764475266288492036980383, 24.81725410963343217162443476672

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