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Properties

Label 1-133-133.97-r0-0-0
Degree $1$
Conductor $133$
Sign $-0.320 + 0.947i$
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  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.173 + 0.984i)6-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.939 + 0.342i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.173 − 0.984i)13-s + (−0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 18-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.173 + 0.984i)6-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.939 + 0.342i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.173 − 0.984i)13-s + (−0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.320 + 0.947i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ -0.320 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9811732946 + 1.367274422i\)
\(L(\frac12)\) \(\approx\) \(0.9811732946 + 1.367274422i\)
\(L(1)\) \(\approx\) \(1.287238727 + 0.9376847251i\)
\(L(1)\) \(\approx\) \(1.287238727 + 0.9376847251i\)

Euler product

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

−28.64552867684454750476602889951, −27.71511310906167551939509301287, −25.997649855017475389015027406971, −25.00157283512779833671802051332, −24.06729007397756603156265321258, −23.40708126586267414845079194662, −22.69473254159087069649720976123, −21.00759461646443313776904843663, −20.39176578428396152917654157739, −19.31729099335306322236692909800, −18.612933474977615897685758264490, −16.96933657503320072524938087479, −15.82961906018297138529052334105, −14.67565242533356469934403735053, −13.696113821444793655968922416469, −12.5075022407988217251972212330, −12.13940194734807199335456223767, −10.92245131463126297682462708034, −9.2076083865518492799118704780, −7.71949529727761139757326021546, −6.847301747131402246774848866390, −5.37987403721695168387508296243, −4.18269620305995478146472801755, −2.711331166350247118529630872923, −1.2965377238806897661663376027, 3.01466228556612618260342345069, 3.51013100623401314860008271989, 4.952378883761300750218789252723, 5.989117354072191286530248912076, 7.58710113351814689207553949363, 8.47874433608028178162433445797, 10.425156806321141143925245720513, 11.08829671782084530644802018892, 12.29396892668518741999477282006, 13.72752139186988794246246425541, 14.666001270018119651918173886766, 15.549981230867467386430210780601, 16.13837339977511919304850402677, 17.38792176852922965710341433489, 19.078279584908221307843759761087, 20.12746347101171007830494311068, 21.18214182519604905001087924655, 21.92434264838511211296518267206, 22.98642804909007011342513015283, 23.530912212257707313597877912416, 25.02013924562635258014787283975, 25.92911561682822983631008594564, 26.83461247612171187232242245335, 27.64918034125815680788746538533, 29.093737915453706714109871186357

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