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Properties

Label 1-133-133.66-r1-0-0
Degree $1$
Conductor $133$
Sign $-0.666 - 0.745i$
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  + (0.173 − 0.984i)2-s + (0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)5-s + (−0.173 − 0.984i)6-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.173 − 0.984i)10-s + 11-s − 12-s + (−0.173 − 0.984i)13-s + (0.766 − 0.642i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)5-s + (−0.173 − 0.984i)6-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.173 − 0.984i)10-s + 11-s − 12-s + (−0.173 − 0.984i)13-s + (0.766 − 0.642i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.666 - 0.745i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ -0.666 - 0.745i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.171005038 - 2.619785699i\)
\(L(\frac12)\) \(\approx\) \(1.171005038 - 2.619785699i\)
\(L(1)\) \(\approx\) \(1.256389315 - 1.192171047i\)
\(L(1)\) \(\approx\) \(1.256389315 - 1.192171047i\)

Euler product

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

−28.591547213369963543871640205952, −27.30242878847612579432942340096, −26.37668148342513587236958198132, −25.838488216790509774801435229762, −24.78671133269585745095452903085, −24.23951113490352748503129850297, −22.55136197995456933551504962950, −21.83742002500143003939390763100, −20.99413400750039798716289139562, −19.50405514158098140219933516253, −18.52496768011723879815995557525, −17.315106191591905567846169586, −16.445155963785658331947364389, −15.15960620980502869869174726821, −14.31903435985355754989207682194, −13.75775918350324900850567483895, −12.55265854744300764724469746439, −10.55135135732696829960749321162, −9.25237050828359736457562779314, −8.79714232612588981912345421180, −7.17450746859456046248785524361, −6.31184411834159045350518182677, −4.75041957609649298804702525045, −3.61329230566140134956522281377, −1.94117192433600875732394275135, 1.066808623760568086328397761312, 2.20633828517286931137362111858, 3.38690615133825058257874393736, 4.81125637250385151306059179393, 6.299455226477831668399522918606, 8.05006999142713768822825375605, 9.30938690492284725285149115222, 9.74494692005333789406549317139, 11.32847237440580374475955752875, 12.65154843468863202307791002535, 13.39226879988027596857322063262, 14.19922176317816953165378252951, 15.27424205301438505790256589620, 17.230230419519275183482377904567, 18.0161671762874505053027133324, 19.15005984189746264550737574012, 20.12120294011610503094064694282, 20.71542367311596949263751866823, 21.80109158435723016096062188975, 22.6759949003353476429301179613, 24.23488042608214527169282628602, 24.95650931515798252822782592858, 26.033008292816447888018933087304, 27.17001084493331729139309360797, 28.0728584067016729840863367458

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