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Properties

Label 1-240-240.173-r0-0-0
Degree $1$
Conductor $240$
Sign $0.160 + 0.987i$
Analytic cond. $1.11455$
Root an. cond. $1.11455$
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  + i·7-s + i·11-s − 13-s + i·17-s − i·19-s + i·23-s + i·29-s + 31-s − 37-s + 41-s + 43-s + i·47-s − 49-s − 53-s − i·59-s + ⋯
L(s)  = 1  + i·7-s + i·11-s − 13-s + i·17-s − i·19-s + i·23-s + i·29-s + 31-s − 37-s + 41-s + 43-s + i·47-s − 49-s − 53-s − i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.160 + 0.987i$
Analytic conductor: \(1.11455\)
Root analytic conductor: \(1.11455\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 240,\ (0:\ ),\ 0.160 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7779974586 + 0.6619231919i\)
\(L(\frac12)\) \(\approx\) \(0.7779974586 + 0.6619231919i\)
\(L(1)\) \(\approx\) \(0.9398727970 + 0.2765568270i\)
\(L(1)\) \(\approx\) \(0.9398727970 + 0.2765568270i\)

Euler product

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

−26.27694313716831214067504686946, −24.775378108028557365592240256252, −24.35836737366173054276682830218, −23.08102906689984220369788622684, −22.48762587832674042193650916485, −21.22710003133663683596402629573, −20.48479559913209620540411750612, −19.44082967317184903779400967968, −18.6532799785335198095108292325, −17.374709536267404744437269794749, −16.66815231526013525031053688038, −15.76304829094414879510587218518, −14.32484926466040086258934702514, −13.85573289863894248049945769372, −12.61298643298470873549324807024, −11.55830070359049145791310703468, −10.498666867120920843407140396771, −9.64428539337236868502041594556, −8.29284512578200698547194900550, −7.35816250438037665328617016271, −6.25268120978319289945514510708, −4.937601191907071202438777508735, −3.816459404169639044327980366796, −2.531852972823107683238800694989, −0.73734661626703111282374998275, 1.81621168787584073550307793274, 2.90978271348126433483145217539, 4.49899425401802655823752884321, 5.461902432762726101869308030813, 6.71986897470070597402540468338, 7.79386118752900662826514429494, 9.02435303858899008156216271116, 9.82196965709380468024791455763, 11.07580547390285887188592324820, 12.24087153091555569991707007800, 12.7865583102077099250989329352, 14.23881989891650826041391055868, 15.164169619363698516264806967576, 15.80035045947763274633715059554, 17.3316096311499113836906407482, 17.765044856108526142351836795868, 19.127631160410463356786763281512, 19.71128802180286809459664477119, 20.963803539947893871511576931667, 21.850929280685467527415474983832, 22.5406966434422838014432475930, 23.76103868239186191640049160652, 24.54819004099251108168278654191, 25.5445083816088713881409836052, 26.20113989833723019321734090302

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