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Properties

Label 1-133-133.40-r0-0-0
Degree $1$
Conductor $133$
Sign $0.197 - 0.980i$
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 + (−0.939 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.197 - 0.980i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ 0.197 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6382981933 - 0.5223448116i\)
\(L(\frac12)\) \(\approx\) \(0.6382981933 - 0.5223448116i\)
\(L(1)\) \(\approx\) \(0.7294783105 - 0.4015675863i\)
\(L(1)\) \(\approx\) \(0.7294783105 - 0.4015675863i\)

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

−28.71563535142825053757816770709, −27.7162448904854924460793966592, −26.90637653762930961605119461543, −25.78082707697511092492288966149, −24.7841593309251989807230105290, −23.986726683486001726606762161790, −22.91692684701432799910658578446, −21.973922563894926007904458686328, −21.2600276554768747393565511991, −19.54374498243782580503568081352, −18.10462617320873508178420596819, −17.52783699108516681015279436231, −16.594492814076622092071281779311, −15.89851011406034006149650791222, −14.48097349274025524331570611465, −13.56156597234589432647537241590, −12.27686190499678606288076386991, −10.88931583862626166874953111684, −9.509108331884716293533591453, −9.024488166905981946685946315637, −7.01270800069267473034082450374, −6.253379200078759024562459025763, −5.1729806980909544670991139750, −4.152459024623181496676736732123, −1.341284205141706982521574228124, 1.17236279518358204759769137039, 2.493188267514203638399119448719, 4.24849838502317705030115372696, 5.617740694692508513036909905241, 6.70481691737824516514053897161, 8.41158618244545122980361451408, 9.81071964454812059151182760939, 10.628739770909649059741684912650, 11.56949475064326915081729911860, 12.755940505840872693703813237988, 13.46373015039253196117244957156, 14.79439903670054779092205011820, 16.68383423792677072326066956367, 17.51222058545061010952944673219, 18.14288763357926239078000918256, 19.19523225028471119917731952940, 20.35621258908269283580861434545, 21.60166938348175851538584386933, 22.274679290522649652186005473517, 22.94468471594327806377857071251, 24.387089967263744857791456240266, 25.44121319792554720568524461550, 26.74306519899129909767732587492, 27.71424455074877415035016005668, 28.58288502957076361406581551380

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