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Properties

Label 2-201-67.66-c2-0-21
Degree $2$
Conductor $201$
Sign $0.0463 - 0.998i$
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  − 3.81i·2-s − 1.73i·3-s − 10.5·4-s − 6.86i·5-s − 6.60·6-s + 3.56i·7-s + 24.8i·8-s − 2.99·9-s − 26.1·10-s + 4.73i·11-s + 18.2i·12-s − 20.4i·13-s + 13.6·14-s − 11.8·15-s + 52.7·16-s + 20.9·17-s + ⋯
L(s)  = 1  − 1.90i·2-s − 0.577i·3-s − 2.63·4-s − 1.37i·5-s − 1.10·6-s + 0.509i·7-s + 3.11i·8-s − 0.333·9-s − 2.61·10-s + 0.430i·11-s + 1.51i·12-s − 1.56i·13-s + 0.971·14-s − 0.793·15-s + 3.29·16-s + 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0463 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0463 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.0463 - 0.998i$
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: \((2,\ 201,\ (\ :1),\ 0.0463 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.683844 + 0.652843i\)
\(L(\frac12)\) \(\approx\) \(0.683844 + 0.652843i\)
\(L(2)\) 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 \( 1 + (66.9 + 3.10i)T \)
good2 \( 1 + 3.81iT - 4T^{2} \)
5 \( 1 + 6.86iT - 25T^{2} \)
7 \( 1 - 3.56iT - 49T^{2} \)
11 \( 1 - 4.73iT - 121T^{2} \)
13 \( 1 + 20.4iT - 169T^{2} \)
17 \( 1 - 20.9T + 289T^{2} \)
19 \( 1 + 13.7T + 361T^{2} \)
23 \( 1 + 26.1T + 529T^{2} \)
29 \( 1 - 47.0T + 841T^{2} \)
31 \( 1 + 1.48iT - 961T^{2} \)
37 \( 1 + 52.3T + 1.36e3T^{2} \)
41 \( 1 - 17.3iT - 1.68e3T^{2} \)
43 \( 1 + 28.1iT - 1.84e3T^{2} \)
47 \( 1 + 75.7T + 2.20e3T^{2} \)
53 \( 1 + 97.4iT - 2.80e3T^{2} \)
59 \( 1 + 15.2T + 3.48e3T^{2} \)
61 \( 1 - 37.9iT - 3.72e3T^{2} \)
71 \( 1 - 4.45T + 5.04e3T^{2} \)
73 \( 1 + 3.98T + 5.32e3T^{2} \)
79 \( 1 + 55.5iT - 6.24e3T^{2} \)
83 \( 1 - 99.4T + 6.88e3T^{2} \)
89 \( 1 - 91.0T + 7.92e3T^{2} \)
97 \( 1 + 36.9iT - 9.40e3T^{2} \)
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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

−12.02011274295899749044253430077, −10.47960048609642197683258997423, −9.798851424501917505118924806025, −8.563507926803705033238778587559, −8.137019201032430801677810907361, −5.61819995191351666509563414828, −4.75072372672621075491060113286, −3.25604613028493674057147049643, −1.82352016202242235073187332920, −0.56650302045876479141048225923, 3.55879015553616924202853209216, 4.61432593120251070569921538173, 6.08256565386164790658863870255, 6.70641827717536422439700195410, 7.66994909413361793945522302517, 8.715317920096164620255864458539, 9.846721450442995390758316148239, 10.60294556928041480757509696423, 12.06380560340418528125467007033, 13.84382719570614200297772404940

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