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Properties

Label 2-201-67.66-c4-0-17
Degree $2$
Conductor $201$
Sign $0.661 + 0.749i$
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  − 4.92i·2-s − 5.19i·3-s − 8.24·4-s + 24.9i·5-s − 25.5·6-s + 17.9i·7-s − 38.1i·8-s − 27·9-s + 123.·10-s + 173. i·11-s + 42.8i·12-s − 5.35i·13-s + 88.5·14-s + 129.·15-s − 319.·16-s + 135.·17-s + ⋯
L(s)  = 1  − 1.23i·2-s − 0.577i·3-s − 0.515·4-s + 0.999i·5-s − 0.710·6-s + 0.366i·7-s − 0.596i·8-s − 0.333·9-s + 1.23·10-s + 1.43i·11-s + 0.297i·12-s − 0.0316i·13-s + 0.451·14-s + 0.576·15-s − 1.24·16-s + 0.470·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.661 + 0.749i$
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: \((2,\ 201,\ (\ :2),\ 0.661 + 0.749i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.967677644\)
\(L(\frac12)\) \(\approx\) \(1.967677644\)
\(L(3)\) 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 \( 1 + (3.36e3 - 2.97e3i)T \)
good2 \( 1 + 4.92iT - 16T^{2} \)
5 \( 1 - 24.9iT - 625T^{2} \)
7 \( 1 - 17.9iT - 2.40e3T^{2} \)
11 \( 1 - 173. iT - 1.46e4T^{2} \)
13 \( 1 + 5.35iT - 2.85e4T^{2} \)
17 \( 1 - 135.T + 8.35e4T^{2} \)
19 \( 1 - 482.T + 1.30e5T^{2} \)
23 \( 1 - 748.T + 2.79e5T^{2} \)
29 \( 1 - 861.T + 7.07e5T^{2} \)
31 \( 1 - 1.13e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.71e3T + 1.87e6T^{2} \)
41 \( 1 + 2.67e3iT - 2.82e6T^{2} \)
43 \( 1 - 820. iT - 3.41e6T^{2} \)
47 \( 1 + 3.74e3T + 4.87e6T^{2} \)
53 \( 1 - 1.51e3iT - 7.89e6T^{2} \)
59 \( 1 - 6.20e3T + 1.21e7T^{2} \)
61 \( 1 - 4.90e3iT - 1.38e7T^{2} \)
71 \( 1 - 5.43e3T + 2.54e7T^{2} \)
73 \( 1 + 4.42e3T + 2.83e7T^{2} \)
79 \( 1 + 5.87e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.18e4T + 4.74e7T^{2} \)
89 \( 1 - 2.55e3T + 6.27e7T^{2} \)
97 \( 1 - 1.26e4iT - 8.85e7T^{2} \)
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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

−11.84472160623392994824975398046, −10.69269746833850737560762391926, −10.05763011244917382557604598952, −8.958871735589001608501082619571, −7.27021736531398549093166815277, −6.81410912041314332245208955281, −5.08213897383485741610095337491, −3.35741538797434969595348507999, −2.50203260963697724358593428248, −1.25104864451405087033618008343, 0.816884291244519445111766475641, 3.26314410632184243994412961761, 4.87218312723942445082179388628, 5.50647735397005856051742377189, 6.69923761416628815450391368654, 7.989386131793008810938392718177, 8.647771157219808643948887521996, 9.609366010834750549318353264597, 10.98483119612754128630062443163, 11.79659327752887566861619850278

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