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Properties

Label 1-133-133.51-r1-0-0
Degree $1$
Conductor $133$
Sign $0.689 - 0.724i$
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  + (0.939 − 0.342i)2-s + (0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (0.766 − 0.642i)6-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.939 + 0.342i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (0.766 − 0.642i)6-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.939 + 0.342i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.766 + 0.642i)13-s + (0.939 + 0.342i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.689 - 0.724i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ 0.689 - 0.724i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.396942038 - 1.884816349i\)
\(L(\frac12)\) \(\approx\) \(4.396942038 - 1.884816349i\)
\(L(1)\) \(\approx\) \(2.606521070 - 0.7674506063i\)
\(L(1)\) \(\approx\) \(2.606521070 - 0.7674506063i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−28.700292557842956482939486567768, −27.35219380394095754848712911688, −26.110100663703892524062956168301, −25.2638070569774094161198009970, −24.79248188045310057574393021266, −23.61516965817361689416653501726, −22.318895581231938619680020854276, −21.46906666140184374124718785705, −20.46036393216262085961333184845, −20.104791156813409818235342913927, −18.28136323705501222716030647251, −16.93350834334293473504140690104, −16.027665841920917456720458424020, −14.872598060847841766747392152373, −14.19135246733596733227246906897, −12.99057908402424211408514993684, −12.4488698926052735007754573685, −10.51341419382185472485442334249, −9.456164961686207823222622413221, −8.116177205910942930705019466834, −7.09466352718307328233635969582, −5.35923402137197766576236144668, −4.63099311484986223303619791411, −3.05754609802679102556665714701, −1.9731311436377011951471901088, 1.6173669787445539903251688579, 2.7121465092426586816264035234, 3.70869601444124139826657318744, 5.441391701079203077112837412140, 6.597485925628162145687986487796, 7.725556823480303303973125994233, 9.43596092934364096634249401935, 10.37382190029123890686209403454, 11.705392032531373825710839511764, 13.021460875286164955370535124, 13.76512247987936166405905631664, 14.547053538556181992293331430195, 15.41847165511447289295975325774, 16.939171881426407397246698644407, 18.63048392696723006282299117636, 19.137903440098117387776447275205, 20.34187050623011293285610343151, 21.49191246723851152318102977824, 21.77612694661407296598261055617, 23.37850802199142735720085629456, 24.19324094833581814127850174389, 25.19553821628917058433174786082, 25.98792778415947462930222497767, 27.05374373716900865587882393194, 28.76244595809753595028226147548

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