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Properties

Label 1-129-129.47-r1-0-0
Degree $1$
Conductor $129$
Sign $-0.367 + 0.930i$
Analytic cond. $13.8629$
Root an. cond. $13.8629$
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  + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + 7-s + (−0.623 − 0.781i)8-s + (0.623 − 0.781i)10-s + (0.900 + 0.433i)11-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)14-s + (0.623 − 0.781i)16-s + (−0.623 + 0.781i)17-s + (−0.900 + 0.433i)19-s + (0.900 + 0.433i)20-s + (−0.222 + 0.974i)22-s + (0.900 + 0.433i)23-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + 7-s + (−0.623 − 0.781i)8-s + (0.623 − 0.781i)10-s + (0.900 + 0.433i)11-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)14-s + (0.623 − 0.781i)16-s + (−0.623 + 0.781i)17-s + (−0.900 + 0.433i)19-s + (0.900 + 0.433i)20-s + (−0.222 + 0.974i)22-s + (0.900 + 0.433i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.367 + 0.930i$
Analytic conductor: \(13.8629\)
Root analytic conductor: \(13.8629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 129,\ (1:\ ),\ -0.367 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9468819185 + 1.391664022i\)
\(L(\frac12)\) \(\approx\) \(0.9468819185 + 1.391664022i\)
\(L(1)\) \(\approx\) \(0.9711074062 + 0.6071877103i\)
\(L(1)\) \(\approx\) \(0.9711074062 + 0.6071877103i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
7 \( 1 + T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 + (-0.900 + 0.433i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + T \)
41 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + (-0.623 + 0.781i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−28.09933799526702094218228034955, −27.24220757888507534379418164690, −26.78984571570645597356779838537, −25.131435143068707871975348482595, −23.88490082404657527645923798137, −22.96706291187240083967470491455, −22.1395651354224890541724211740, −21.10511443651084092327140621551, −20.09746649169313616641545607889, −19.16001019558784977858173232665, −18.22116367616678706799942370528, −17.31890194258726388566697567709, −15.46883933582612722491100025559, −14.56262499138277858567160884693, −13.65675198794100471043859298493, −12.25851696151181101287536507044, −11.1311104503531246568657273859, −10.81403953279964104437496313913, −9.09167658873631090910332196187, −8.05218693313027041552599465951, −6.43049059891996891850854455553, −4.82309944889844608848979950059, −3.712050716218911591051409266570, −2.43934299417611800574352266510, −0.73868800949571264002000710141, 1.364641112537180818820572609877, 4.01490061209106239451595753734, 4.607673945520787602418792721391, 6.07479317588579858013856665816, 7.364612388979033464249054850479, 8.48305727464534914971320068646, 9.157379767968416250914982660009, 11.15295271280530188955478511939, 12.27063566985254770847494054433, 13.33943053178543452235059019361, 14.62475632707031101709502238804, 15.28707984158492747934885356002, 16.63769046432993166609012042029, 17.18123615456885333106605651280, 18.4190582968160284409312860813, 19.656808318882820423676740424355, 20.92844574471130329050835991374, 21.83227897446560038608814903006, 23.256555404927721505102816056438, 23.81697053205789286101142444724, 24.71142161629663608592833832105, 25.60145841067736197818525856926, 26.915379806842864278813384447301, 27.61176058430116059313035004533, 28.40656226154638651270035362262

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