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Properties

Label 1-133-133.131-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: $\chi_{133} (131, \cdot )$
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

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

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