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Properties

Label 1-12e3-1728.859-r1-0-0
Degree $1$
Conductor $1728$
Sign $-0.833 + 0.552i$
Analytic cond. $185.699$
Root an. cond. $185.699$
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  + (−0.953 + 0.300i)5-s + (−0.906 − 0.422i)7-s + (−0.887 + 0.461i)11-s + (−0.537 − 0.843i)13-s + (−0.866 + 0.5i)17-s + (0.130 − 0.991i)19-s + (0.906 − 0.422i)23-s + (0.819 − 0.573i)25-s + (0.976 − 0.216i)29-s + (−0.939 − 0.342i)31-s + (0.991 + 0.130i)35-s + (−0.130 − 0.991i)37-s + (0.819 + 0.573i)41-s + (0.887 − 0.461i)43-s + (−0.342 − 0.939i)47-s + ⋯
L(s)  = 1  + (−0.953 + 0.300i)5-s + (−0.906 − 0.422i)7-s + (−0.887 + 0.461i)11-s + (−0.537 − 0.843i)13-s + (−0.866 + 0.5i)17-s + (0.130 − 0.991i)19-s + (0.906 − 0.422i)23-s + (0.819 − 0.573i)25-s + (0.976 − 0.216i)29-s + (−0.939 − 0.342i)31-s + (0.991 + 0.130i)35-s + (−0.130 − 0.991i)37-s + (0.819 + 0.573i)41-s + (0.887 − 0.461i)43-s + (−0.342 − 0.939i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.833 + 0.552i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ -0.833 + 0.552i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07558992232 - 0.2508272933i\)
\(L(\frac12)\) \(\approx\) \(-0.07558992232 - 0.2508272933i\)
\(L(1)\) \(\approx\) \(0.6486128066 - 0.1124510790i\)
\(L(1)\) \(\approx\) \(0.6486128066 - 0.1124510790i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.953 + 0.300i)T \)
7 \( 1 + (-0.906 - 0.422i)T \)
11 \( 1 + (-0.887 + 0.461i)T \)
13 \( 1 + (-0.537 - 0.843i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.130 - 0.991i)T \)
23 \( 1 + (0.906 - 0.422i)T \)
29 \( 1 + (0.976 - 0.216i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (-0.130 - 0.991i)T \)
41 \( 1 + (0.819 + 0.573i)T \)
43 \( 1 + (0.887 - 0.461i)T \)
47 \( 1 + (-0.342 - 0.939i)T \)
53 \( 1 + (0.382 - 0.923i)T \)
59 \( 1 + (0.953 - 0.300i)T \)
61 \( 1 + (0.675 + 0.737i)T \)
67 \( 1 + (-0.843 + 0.537i)T \)
71 \( 1 + (0.258 - 0.965i)T \)
73 \( 1 + (-0.258 - 0.965i)T \)
79 \( 1 + (-0.984 - 0.173i)T \)
83 \( 1 + (0.216 + 0.976i)T \)
89 \( 1 + (-0.965 + 0.258i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−20.4769828349488144485953815812, −19.6890890460826892476448295070, −19.02926032425885991396443357365, −18.64705786911187210184694204111, −17.597532177450826892124895394734, −16.57278425678060846358855583958, −16.05125052900051760961138830710, −15.60686584155963434621839527553, −14.684244747398825384951355730912, −13.801461903819626364359562071299, −12.880253468828989515651893706781, −12.405229170227385767012653525276, −11.555198740431960583286799709535, −10.88305756568754171788960349888, −9.88054163280519341065091646558, −9.06360745650512741781391676406, −8.457944869982302359011629556678, −7.45109928975854259728036351295, −6.86251757472435790991507530377, −5.822926753008009708057815976244, −4.95907731248049961493926266383, −4.12672456821782116618207780407, −3.16680167686457568689399839454, −2.49138638553542728227372070717, −1.077441079430289524948933667479, 0.084465046398712230125135470758, 0.61818106406370054981264986586, 2.41345291059876363735809379180, 2.97002998089309684770533461116, 3.96894268004296132297251455473, 4.71322215093121119792964039756, 5.68227116064732303659216508113, 6.9186594359692010159212376646, 7.175748840654752179090342241694, 8.12592640693358041541935482021, 8.96343639666085022757698924655, 9.95621146519342916850456077718, 10.66641639495883455261314439069, 11.20104380902817727409451827059, 12.37796623641203748593138015251, 12.86372319745186492181161801544, 13.48524469225459976928624552857, 14.742787057543326347870788986874, 15.2418760374842305320746193041, 15.906786084249390448323633405247, 16.533326655084155316130405390465, 17.64352587001044029255383208072, 18.06367093296202211914399727666, 19.26960781476225527000515600318, 19.54122773504640704282724353990

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