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Properties

Label 1-133-133.65-r1-0-0
Degree $1$
Conductor $133$
Sign $0.632 - 0.774i$
Analytic cond. $14.2928$
Root an. cond. $14.2928$
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  + (0.5 − 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s − 8-s + 9-s + (0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + 17-s + (0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s − 8-s + 9-s + (0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + 17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.632 - 0.774i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ 0.632 - 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.193350623 - 0.5661348086i\)
\(L(\frac12)\) \(\approx\) \(1.193350623 - 0.5661348086i\)
\(L(1)\) \(\approx\) \(0.8823892522 - 0.3344844245i\)
\(L(1)\) \(\approx\) \(0.8823892522 - 0.3344844245i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 - T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−28.45402813756659202116774843567, −27.408535818333757504726505698889, −26.64485949747518482662163405149, −25.238566375432890478574670318943, −24.219376763241904229154332182868, −23.55844506382312280182075567368, −22.946467277434501013009196604602, −21.47543287184456683344658862824, −21.06648676525870476097825218873, −19.14626743536249100310200595991, −18.10069323828840752789904564136, −16.817229562378688975685952476283, −16.35461352620353696534681668436, −15.54250045473040085215608388159, −14.015722732825499969226649917011, −12.87903997395265213107322164809, −12.085910584361332420836931940189, −10.99180769997466922610091413783, −9.20512563658750413945037321986, −8.07767414783507076173767399200, −6.86460393841499783738573909123, −5.64249438184869858411585910265, −4.82785050995256104624576628990, −3.60277720859134787685162608160, −0.816176270619748882453453531032, 0.83922939491107761821493602916, 2.68542270959926767463890434241, 4.04223125780093409111207662951, 5.263610439771659008778829179575, 6.39876797299689962368499999536, 7.76871053194245257133586338291, 9.875425027173163068567231584, 10.55592109892061222976790304101, 11.516929530604065715004659085331, 12.39912415853744375188484270162, 13.432427188831044523874266416786, 14.96113505991342343535607584341, 15.59262462463318709072872860986, 17.291227516479126830415642613966, 18.36078134953668941819147261873, 18.938507000595602702664151056, 20.33573717714886328423881792171, 21.325587444543599713918391454465, 22.416504888526401832598637097084, 23.110720834342351635089030882103, 23.545491081618536927346587959370, 25.10738845573007667561439070633, 26.64055428119717012044651869922, 27.61204485769358875251733372117, 28.25467571416186858565344846941

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