Properties

Label 2-201-67.66-c2-0-3
Degree $2$
Conductor $201$
Sign $0.997 - 0.0745i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.60i·2-s + 1.73i·3-s − 9.02·4-s + 2.97i·5-s + 6.25·6-s + 2.36i·7-s + 18.1i·8-s − 2.99·9-s + 10.7·10-s + 15.8i·11-s − 15.6i·12-s + 6.13i·13-s + 8.51·14-s − 5.14·15-s + 29.2·16-s + 11.1·17-s + ⋯
L(s)  = 1  − 1.80i·2-s + 0.577i·3-s − 2.25·4-s + 0.594i·5-s + 1.04·6-s + 0.337i·7-s + 2.26i·8-s − 0.333·9-s + 1.07·10-s + 1.44i·11-s − 1.30i·12-s + 0.471i·13-s + 0.608·14-s − 0.343·15-s + 1.83·16-s + 0.658·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.997 - 0.0745i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.997 - 0.0745i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05060 + 0.0392172i\)
\(L(\frac12)\) \(\approx\) \(1.05060 + 0.0392172i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
67 \( 1 + (-4.99 - 66.8i)T \)
good2 \( 1 + 3.60iT - 4T^{2} \)
5 \( 1 - 2.97iT - 25T^{2} \)
7 \( 1 - 2.36iT - 49T^{2} \)
11 \( 1 - 15.8iT - 121T^{2} \)
13 \( 1 - 6.13iT - 169T^{2} \)
17 \( 1 - 11.1T + 289T^{2} \)
19 \( 1 - 18.6T + 361T^{2} \)
23 \( 1 + 24.0T + 529T^{2} \)
29 \( 1 + 48.8T + 841T^{2} \)
31 \( 1 - 34.1iT - 961T^{2} \)
37 \( 1 + 21.5T + 1.36e3T^{2} \)
41 \( 1 - 0.389iT - 1.68e3T^{2} \)
43 \( 1 - 6.25iT - 1.84e3T^{2} \)
47 \( 1 + 9.41T + 2.20e3T^{2} \)
53 \( 1 - 52.4iT - 2.80e3T^{2} \)
59 \( 1 - 70.4T + 3.48e3T^{2} \)
61 \( 1 + 43.4iT - 3.72e3T^{2} \)
71 \( 1 - 8.27T + 5.04e3T^{2} \)
73 \( 1 - 28.8T + 5.32e3T^{2} \)
79 \( 1 + 107. iT - 6.24e3T^{2} \)
83 \( 1 + 151.T + 6.88e3T^{2} \)
89 \( 1 + 99.8T + 7.92e3T^{2} \)
97 \( 1 + 164. iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.03440044328843391589219654827, −11.28438494479409472494632712937, −10.24468568825692842314703082426, −9.759918560614393520415783344282, −8.837563579884172843911052657216, −7.28469068457548950847399373469, −5.37413043003085715432765827456, −4.22214917725348375153680869316, −3.13173891748241004384590706640, −1.85806801482757117951887927512, 0.62873406673255338293836058152, 3.72101641336850145849886060343, 5.36575457557065269554920379871, 5.89424182825535122395812288439, 7.17735430260201995833604391744, 7.993067749516923517575149221255, 8.691933946230182880773359183525, 9.785475512725015144055630120578, 11.36038315744147098326268731366, 12.67594378982155702424651933546

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