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Properties

Label 2-201-67.66-c4-0-38
Degree $2$
Conductor $201$
Sign $-0.621 - 0.783i$
Analytic cond. $20.7773$
Root an. cond. $4.55821$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.06i·2-s − 5.19i·3-s − 9.63·4-s + 45.6i·5-s − 26.3·6-s − 49.9i·7-s − 32.2i·8-s − 27·9-s + 231.·10-s − 197. i·11-s + 50.0i·12-s + 121. i·13-s − 253.·14-s + 237.·15-s − 317.·16-s − 137.·17-s + ⋯
L(s)  = 1  − 1.26i·2-s − 0.577i·3-s − 0.602·4-s + 1.82i·5-s − 0.730·6-s − 1.01i·7-s − 0.503i·8-s − 0.333·9-s + 2.31·10-s − 1.63i·11-s + 0.347i·12-s + 0.717i·13-s − 1.29·14-s + 1.05·15-s − 1.23·16-s − 0.475·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.621 - 0.783i$
Analytic conductor: \(20.7773\)
Root analytic conductor: \(4.55821\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :2),\ -0.621 - 0.783i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.8331642427\)
\(L(\frac12)\) \(\approx\) \(0.8331642427\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19iT \)
67 \( 1 + (-3.51e3 + 2.78e3i)T \)
good2 \( 1 + 5.06iT - 16T^{2} \)
5 \( 1 - 45.6iT - 625T^{2} \)
7 \( 1 + 49.9iT - 2.40e3T^{2} \)
11 \( 1 + 197. iT - 1.46e4T^{2} \)
13 \( 1 - 121. iT - 2.85e4T^{2} \)
17 \( 1 + 137.T + 8.35e4T^{2} \)
19 \( 1 - 98.9T + 1.30e5T^{2} \)
23 \( 1 + 640.T + 2.79e5T^{2} \)
29 \( 1 + 385.T + 7.07e5T^{2} \)
31 \( 1 + 465. iT - 9.23e5T^{2} \)
37 \( 1 + 353.T + 1.87e6T^{2} \)
41 \( 1 - 1.11e3iT - 2.82e6T^{2} \)
43 \( 1 - 332. iT - 3.41e6T^{2} \)
47 \( 1 + 3.69e3T + 4.87e6T^{2} \)
53 \( 1 + 2.42e3iT - 7.89e6T^{2} \)
59 \( 1 - 1.05e3T + 1.21e7T^{2} \)
61 \( 1 + 7.09e3iT - 1.38e7T^{2} \)
71 \( 1 + 8.49e3T + 2.54e7T^{2} \)
73 \( 1 - 2.42e3T + 2.83e7T^{2} \)
79 \( 1 - 8.88e3iT - 3.89e7T^{2} \)
83 \( 1 - 4.84e3T + 4.74e7T^{2} \)
89 \( 1 + 6.24e3T + 6.27e7T^{2} \)
97 \( 1 + 1.07e4iT - 8.85e7T^{2} \)
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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

−11.30691056800323114364938398352, −10.53211770282941871500358017184, −9.655751786682690750629994633009, −8.010086878174361126976326131716, −6.88174848473832415232976855989, −6.25055901342752233866771502880, −3.84906654917877907392337031051, −3.11668200247045045636149522756, −1.90620064188667346590573973172, −0.27237221107278858771979609875, 2.00895440761600185592385323849, 4.41081386284855150836807821701, 5.17041966902558144723867926284, 5.89639685328793419351905484464, 7.47334622561818340509032191552, 8.460090230420149626983157411457, 9.072851739391907863575321456066, 10.01870262135911153367559566311, 11.78233678188581680562113291373, 12.41418108142428972914101117405

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