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Properties

Label 1-135-135.104-r1-0-0
Degree $1$
Conductor $135$
Sign $-0.893 + 0.448i$
Analytic cond. $14.5077$
Root an. cond. $14.5077$
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  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.939 − 0.342i)11-s + (−0.766 + 0.642i)13-s + (−0.766 + 0.642i)14-s + (−0.939 + 0.342i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.939 + 0.342i)22-s + (0.173 + 0.984i)23-s − 26-s − 28-s + (−0.766 − 0.642i)29-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.939 − 0.342i)11-s + (−0.766 + 0.642i)13-s + (−0.766 + 0.642i)14-s + (−0.939 + 0.342i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.939 + 0.342i)22-s + (0.173 + 0.984i)23-s − 26-s − 28-s + (−0.766 − 0.642i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.893 + 0.448i$
Analytic conductor: \(14.5077\)
Root analytic conductor: \(14.5077\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (1:\ ),\ -0.893 + 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4963171779 + 2.094126837i\)
\(L(\frac12)\) \(\approx\) \(0.4963171779 + 2.094126837i\)
\(L(1)\) \(\approx\) \(1.133095829 + 0.9399209549i\)
\(L(1)\) \(\approx\) \(1.133095829 + 0.9399209549i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (0.939 - 0.342i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−28.027902469002063551080936703753, −27.12235562413353914508715713025, −25.88498156065350891856336572444, −24.545256173596679549590432036726, −23.82838759525298120125562942136, −22.61901001931390962493397691824, −22.11760331823383847298515306363, −20.76541503492328230995923056640, −19.891012961829423226243049938546, −19.2931111843094577177368679081, −17.70119541487532359548618214773, −16.69642888803396124682928713797, −15.168996254085331556803317493875, −14.44619164576971377439931247618, −13.22287529022492149969555677864, −12.47390627696760827919818626851, −11.12756726206280043446001825817, −10.303765649026193849108342992917, −9.12794621151529522980012270549, −7.26977525129123372727977851026, −6.21867347776775457475279305251, −4.664181584200685602781115182557, −3.782424763935420470993633398511, −2.2508116295022876473626510607, −0.64019134318378091277318118665, 2.22173311302235178481761845076, 3.62810576117823645306201269676, 4.96229911415734891772758155160, 6.09679911700558641136247035952, 7.10905710256079801543947719189, 8.52393488083948691486034627981, 9.497320459087655292317069352389, 11.543947979090749751640688906858, 12.10605247635041217188435943035, 13.392139593721799755154849165236, 14.4470707535614744065304072897, 15.29116638289779700894982500710, 16.38097775414134817961881940072, 17.258536344998870228921851320889, 18.54737511436443375003198086720, 19.71060452670389183956841588453, 21.10335772416814639788233113893, 21.92573632804805770504762222125, 22.638738626220733469917559148450, 23.85082871342788068006212009369, 24.853546658783447052200357050757, 25.32060093896823256213707824712, 26.6615249144601741069566227720, 27.50952569073379679853562594519, 28.98340498049403409930974168135

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