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Properties

Label 2-201-3.2-c2-0-43
Degree $2$
Conductor $201$
Sign $-0.00691 - 0.999i$
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  − 3.83i·2-s + (−0.0207 − 2.99i)3-s − 10.7·4-s − 4.79i·5-s + (−11.5 + 0.0796i)6-s + 5.67·7-s + 25.8i·8-s + (−8.99 + 0.124i)9-s − 18.3·10-s − 12.6i·11-s + (0.222 + 32.2i)12-s + 1.63·13-s − 21.7i·14-s + (−14.3 + 0.0993i)15-s + 56.3·16-s + 19.9i·17-s + ⋯
L(s)  = 1  − 1.91i·2-s + (−0.00691 − 0.999i)3-s − 2.68·4-s − 0.958i·5-s + (−1.91 + 0.0132i)6-s + 0.810·7-s + 3.23i·8-s + (−0.999 + 0.0138i)9-s − 1.83·10-s − 1.15i·11-s + (0.0185 + 2.68i)12-s + 0.125·13-s − 1.55i·14-s + (−0.958 + 0.00662i)15-s + 3.52·16-s + 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00691 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00691 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.00691 - 0.999i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.00691 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.828034 + 0.833778i\)
\(L(\frac12)\) \(\approx\) \(0.828034 + 0.833778i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0207 + 2.99i)T \)
67 \( 1 - 8.18T \)
good2 \( 1 + 3.83iT - 4T^{2} \)
5 \( 1 + 4.79iT - 25T^{2} \)
7 \( 1 - 5.67T + 49T^{2} \)
11 \( 1 + 12.6iT - 121T^{2} \)
13 \( 1 - 1.63T + 169T^{2} \)
17 \( 1 - 19.9iT - 289T^{2} \)
19 \( 1 - 26.4T + 361T^{2} \)
23 \( 1 + 26.2iT - 529T^{2} \)
29 \( 1 + 34.8iT - 841T^{2} \)
31 \( 1 + 32.9T + 961T^{2} \)
37 \( 1 + 11.9T + 1.36e3T^{2} \)
41 \( 1 - 76.6iT - 1.68e3T^{2} \)
43 \( 1 + 3.69T + 1.84e3T^{2} \)
47 \( 1 - 48.3iT - 2.20e3T^{2} \)
53 \( 1 + 26.7iT - 2.80e3T^{2} \)
59 \( 1 + 110. iT - 3.48e3T^{2} \)
61 \( 1 + 24.2T + 3.72e3T^{2} \)
71 \( 1 - 66.3iT - 5.04e3T^{2} \)
73 \( 1 - 73.3T + 5.32e3T^{2} \)
79 \( 1 + 26.2T + 6.24e3T^{2} \)
83 \( 1 - 19.9iT - 6.88e3T^{2} \)
89 \( 1 + 85.9iT - 7.92e3T^{2} \)
97 \( 1 + 101.T + 9.40e3T^{2} \)
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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

−11.43424711758306607173436598252, −11.04969386642009596487208952115, −9.613239561667671492320444826362, −8.467666970092872270909637056299, −8.157124500677841457204444613563, −5.79738168860256119775182461511, −4.69372056356559880654136061438, −3.22614047602120012924427832109, −1.72230814468754428727245507231, −0.74276858718372782494714467274, 3.55427753483850890439151340972, 4.91621362915815279312060044347, 5.49032055482935754245429267702, 7.07760372707005487426748323593, 7.51526084518028017686805452736, 8.926336905519414921277948661340, 9.617834884333574007132844801400, 10.67850795346413030653156940920, 11.97858198131151267980721437035, 13.68106541308846237210127227612

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