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Properties

Label 1-133-133.13-r0-0-0
Degree $1$
Conductor $133$
Sign $-0.173 + 0.984i$
Analytic cond. $0.617649$
Root an. cond. $0.617649$
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 + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)5-s + (−0.766 + 0.642i)6-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.173 + 0.984i)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 + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s − 18-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)5-s + (−0.766 + 0.642i)6-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.173 + 0.984i)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 + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8332490016 + 0.9923695959i\)
\(L(\frac12)\) \(\approx\) \(0.8332490016 + 0.9923695959i\)
\(L(1)\) \(\approx\) \(0.9972896659 + 0.7269408833i\)
\(L(1)\) \(\approx\) \(0.9972896659 + 0.7269408833i\)

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 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
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

−28.80376032769025174686722969884, −27.32377540620406691132770307054, −26.21430733813166432124152020709, −25.7334590679039441156903040478, −24.39578350455678824194722433934, −23.30529876550614269463741528513, −21.97023076135203417057141856854, −21.13228047135974845209424990882, −20.35369293640928399345919380785, −19.14711687791645927068565835049, −18.37127682903599910062605261658, −17.75292801545352058488938793231, −16.1748675471940680817614790028, −14.27482339745472255381239262304, −13.78134781414590734021431356925, −12.92933563105364407178200664789, −11.620070597578544528781303868998, −10.45400852335916170230687459419, −9.26260113548233449425507722384, −8.50142934029239543177519020958, −7.01750420300624495085598269069, −5.51951286377137653227446383015, −3.62134995717478541636403251430, −2.56875459517185461241414064619, −1.43227536749621714918052629918, 1.988309252742824759954701040974, 3.916377807908551063902171595040, 5.110782883909317995987145657505, 6.19302335403385672610112190301, 7.81332454760109498937150360278, 8.66451082790384042972756644692, 9.79115262480142693472076644457, 10.435109385754387617205822116576, 12.90078668007488581216646168043, 13.56475863630659238114625453754, 14.71728663398982969871396526813, 15.496372380154374578167966723060, 16.53641012055741111210525952238, 17.57070137705866210881781059528, 18.50184071284126183599740006560, 19.91661917316745789674271675838, 20.90135930709383674200206162238, 21.91014275572478933947969023524, 22.92826434571683303520533262552, 24.26892543787586299544617961287, 25.15218045487203147443473426962, 25.92419509725661165301057395808, 26.43674340447317358879311250485, 28.04480918517226100479748551904, 28.21445964595811652639322643783

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