Properties

Label 2-201-67.66-c4-0-0
Degree $2$
Conductor $201$
Sign $0.671 - 0.741i$
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  − 6.42i·2-s + 5.19i·3-s − 25.3·4-s − 17.3i·5-s + 33.3·6-s − 16.4i·7-s + 59.8i·8-s − 27·9-s − 111.·10-s + 167. i·11-s − 131. i·12-s − 43.2i·13-s − 105.·14-s + 90.3·15-s − 20.4·16-s − 462.·17-s + ⋯
L(s)  = 1  − 1.60i·2-s + 0.577i·3-s − 1.58·4-s − 0.695i·5-s + 0.927·6-s − 0.335i·7-s + 0.934i·8-s − 0.333·9-s − 1.11·10-s + 1.38i·11-s − 0.913i·12-s − 0.256i·13-s − 0.539·14-s + 0.401·15-s − 0.0797·16-s − 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.671 - 0.741i$
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: \((2,\ 201,\ (\ :2),\ 0.671 - 0.741i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.3396749757\)
\(L(\frac12)\) \(\approx\) \(0.3396749757\)
\(L(3)\) 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 \( 1 + (3.32e3 + 3.01e3i)T \)
good2 \( 1 + 6.42iT - 16T^{2} \)
5 \( 1 + 17.3iT - 625T^{2} \)
7 \( 1 + 16.4iT - 2.40e3T^{2} \)
11 \( 1 - 167. iT - 1.46e4T^{2} \)
13 \( 1 + 43.2iT - 2.85e4T^{2} \)
17 \( 1 + 462.T + 8.35e4T^{2} \)
19 \( 1 + 300.T + 1.30e5T^{2} \)
23 \( 1 + 132.T + 2.79e5T^{2} \)
29 \( 1 - 698.T + 7.07e5T^{2} \)
31 \( 1 - 1.71e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.69e3T + 1.87e6T^{2} \)
41 \( 1 - 2.00e3iT - 2.82e6T^{2} \)
43 \( 1 - 468. iT - 3.41e6T^{2} \)
47 \( 1 + 2.84e3T + 4.87e6T^{2} \)
53 \( 1 + 2.63e3iT - 7.89e6T^{2} \)
59 \( 1 + 4.08e3T + 1.21e7T^{2} \)
61 \( 1 - 1.82e3iT - 1.38e7T^{2} \)
71 \( 1 + 4.10e3T + 2.54e7T^{2} \)
73 \( 1 + 9.08e3T + 2.83e7T^{2} \)
79 \( 1 - 6.59e3iT - 3.89e7T^{2} \)
83 \( 1 + 6.55e3T + 4.74e7T^{2} \)
89 \( 1 - 4.04e3T + 6.27e7T^{2} \)
97 \( 1 + 612. iT - 8.85e7T^{2} \)
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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

−11.85631042228760986817829250774, −10.83892819190774745583310115012, −10.17241820609656148624221697410, −9.283539451267203514729127492701, −8.452761086500137819171080655887, −6.72906545223601186950513874906, −4.68588611484873853846527741718, −4.39253003300535043362306281497, −2.79676137316175714056751582312, −1.49681188510967080714211641597, 0.11880127437870111780711913517, 2.54503095275694768072433173225, 4.38059557009602912106408999192, 5.95687035351228607612741592933, 6.37260399878158337203048480163, 7.38416265321658291312200781256, 8.430968471897814972685659226913, 9.047627585521481551017562702816, 10.76928178733518860565308133066, 11.59965264233604029907989138201

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