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Properties

Label 1-133-133.68-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: $\chi_{133} (68, \cdot )$
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

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

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