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Properties

Label 1-133-133.34-r0-0-0
Degree $1$
Conductor $133$
Sign $-0.891 + 0.452i$
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  + (−0.766 − 0.642i)2-s + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.939 + 0.342i)6-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.766 − 0.642i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.939 − 0.342i)13-s + (−0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s − 18-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.939 + 0.342i)6-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.766 − 0.642i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.939 − 0.342i)13-s + (−0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.891 + 0.452i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ -0.891 + 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03761672281 + 0.1572966920i\)
\(L(\frac12)\) \(\approx\) \(0.03761672281 + 0.1572966920i\)
\(L(1)\) \(\approx\) \(0.3908886478 + 0.06007692549i\)
\(L(1)\) \(\approx\) \(0.3908886478 + 0.06007692549i\)

Euler product

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

−28.30515736453565423909745831827, −27.22396471633460748224015104536, −26.43954957550907894046584035977, −24.87453671089673768218223260745, −24.23522804561379091584008829987, −23.69629765171723240637554423000, −22.42550053163376947144801699027, −21.1428490611503481237160603449, −19.74645817090110482920176210832, −18.92630137925606563513291031309, −17.80814398807113208364240570382, −16.86384788065726918755557333938, −16.338683065250211169563748297045, −15.24074805190184680421755517870, −13.65194558946828730467874427550, −12.51001683540956891988645977652, −11.31406869380835831377327475139, −10.28023190172205553538077313996, −8.92896480029246778828939235449, −7.91190633966185171431036622171, −6.70327122512905962415864856252, −5.55129086458761998781537284770, −4.63672113692761944995043101464, −1.80531138841483965124159066878, −0.20047668378407859342170346219, 2.14194615900929526786047576376, 3.59635087483551650554503486018, 5.05476719859810746094274986705, 6.849747577570715303160079047569, 7.54176329004699219073704020845, 9.45072968498146343555175599965, 10.23737694479171592160981015319, 11.15475909118743598734681093530, 11.995304899424737530362861940764, 13.13717889995446362553788092705, 15.010384868249136540110724153385, 15.8717836272160911049500059824, 17.21005031746174398779053500669, 17.88675479930209786354169257018, 18.70631263376267695696682319568, 19.91921858151006430398134719960, 21.00180629431200255026908589794, 22.22236950437916247809180643519, 22.53636133122612883121200437957, 23.91895794296218031522423297135, 25.43738896189053053243295991706, 26.41813835416621427636717516201, 27.22797446957770389501381622713, 27.93558674821861137334047529200, 29.157283468298455203354578169690

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