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Properties

Label 1-133-133.45-r1-0-0
Degree $1$
Conductor $133$
Sign $-0.423 + 0.905i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.423 + 0.905i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ -0.423 + 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1882109193 + 0.2958116253i\)
\(L(\frac12)\) \(\approx\) \(0.1882109193 + 0.2958116253i\)
\(L(1)\) \(\approx\) \(0.5428029394 - 0.04685271031i\)
\(L(1)\) \(\approx\) \(0.5428029394 - 0.04685271031i\)

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

−28.12984238376688627229680003287, −27.229049127586741945124837997922, −25.96793099751048279574305799518, −24.94484269697282134940203171163, −24.19881097556283149710742513640, −23.19690090268204601610230772771, −22.43054332435789434414991366787, −21.02829056644031258332491935375, −19.90917048203220538774133669850, −18.3568485443963088493608392425, −17.69763154656316300202330761919, −16.93614852300866155170125020080, −15.93264331240527253719977547097, −15.124536073091347591870869855276, −13.35496474397329624019007605881, −12.71097411678691957143646640459, −11.00918391907402248841756659868, −10.01519018801658127140574029488, −8.9536984746196772713114257072, −7.612903123756812295439119089099, −6.39888247200410225371625785407, −5.34477154271739479050483821166, −4.5713534944895121538014768065, −1.592565194949391641394365157530, −0.19637832836682717354639804747, 1.539688696189424856649517723505, 3.0424407718918226352421625341, 4.55544787246482537479277012540, 6.10766318393778311684341948491, 7.20735127578674223738560607910, 8.83225347805978822269566776093, 10.074785784700686233846738258802, 11.00982416352676283806038952256, 11.52147439044831383659189510084, 13.0226398522766263808845681496, 13.83581928970773259510397452139, 15.64506235704219167484713796000, 16.80792878974390436751781154948, 17.69566975082330347631198868116, 18.58452136798467792922917046279, 19.20302816751629882407089576163, 20.98334528190556115490784418012, 21.57968049750073842910275528870, 22.463395251911877489082327950336, 23.35859489040467720795430289799, 24.73133115430005655619742596823, 26.27216652292772331451202660712, 26.701581397479734322504665972064, 27.87757301786118933293741424237, 29.03497337703892162751760604752

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