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Properties

Label 1-133-133.18-r1-0-0
Degree $1$
Conductor $133$
Sign $-0.832 + 0.553i$
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 + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s − 13-s − 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s − 13-s − 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ -0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09653025218 + 0.3194099816i\)
\(L(\frac12)\) \(\approx\) \(0.09653025218 + 0.3194099816i\)
\(L(1)\) \(\approx\) \(0.9417636792 - 0.05976449703i\)
\(L(1)\) \(\approx\) \(0.9417636792 - 0.05976449703i\)

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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 - T \)
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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

−27.91417373570117030904753054846, −26.57877462452322237702632165452, −25.81022440611469094495971247149, −24.67026301474857590515129927160, −24.170205198878378073496790354360, −23.35481041045445605208646556072, −22.25005774636940100374716388163, −20.78257097413468601001101146212, −20.00858280538785039522955515121, −18.74880700973279342652393029057, −17.55726521274223838705068297617, −16.77547383350915471746854998207, −15.36568713373886668456513756042, −14.71567203982342551911943894332, −13.31287859905869289270441686704, −12.66728784890774129319787208154, −11.857095199215507689793803024165, −9.56421029276816737757867088534, −8.3408136914356170052299343556, −7.67298748632742929111636101601, −6.550596185014505596474794722654, −5.10330779475279613319717834684, −3.945763674654099728891255920080, −2.24419676786033872326168164843, −0.097158896669715965406183297151, 2.46786120392667554596999870502, 3.28739475908482988688115498665, 4.45162553315376443625757593577, 5.68616121895638245341527892885, 7.51953193751415652081828612248, 8.99869996837167887029230995674, 10.09245200578679424203431712312, 10.96434897362734369494531323170, 11.82904317709200369143184101503, 13.464963566679687247946910113303, 14.30556536678002903776477889486, 15.19569929296358808516930274537, 16.1262010453813910939799730961, 17.9047513360325078322403420565, 19.116098017370281158465240632370, 19.72598193082967694852689210804, 20.81864967803110859432355855959, 21.84880684466230696420145239305, 22.35907489429860527426767696994, 23.4839028799501883210680921501, 24.67353171527622247005963964727, 26.224452661215452568223751893844, 26.968627489641767936117395804866, 27.59794905647810120430442773426, 28.927572320506510096864966320975

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