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Properties

Label 1-133-133.37-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: $\chi_{133} (37, \cdot )$
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

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

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