Properties

Label 2-201-201.38-c1-0-12
Degree $2$
Conductor $201$
Sign $0.888 - 0.458i$
Analytic cond. $1.60499$
Root an. cond. $1.26688$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.314 + 0.544i)2-s + (1.35 − 1.08i)3-s + (0.802 + 1.38i)4-s + 3.91·5-s + (0.164 + 1.07i)6-s + (−3.42 + 1.97i)7-s − 2.26·8-s + (0.653 − 2.92i)9-s + (−1.22 + 2.12i)10-s + (0.0182 + 0.0316i)11-s + (2.59 + 1.00i)12-s + (−4.23 − 2.44i)13-s − 2.48i·14-s + (5.28 − 4.23i)15-s + (−0.892 + 1.54i)16-s + (−2.19 − 1.26i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.384i)2-s + (0.780 − 0.625i)3-s + (0.401 + 0.694i)4-s + 1.74·5-s + (0.0673 + 0.439i)6-s + (−1.29 + 0.747i)7-s − 0.801·8-s + (0.217 − 0.976i)9-s + (−0.388 + 0.673i)10-s + (0.00550 + 0.00953i)11-s + (0.747 + 0.291i)12-s + (−1.17 − 0.677i)13-s − 0.664i·14-s + (1.36 − 1.09i)15-s + (−0.223 + 0.386i)16-s + (−0.533 − 0.307i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53239 + 0.372006i\)
\(L(\frac12)\) \(\approx\) \(1.53239 + 0.372006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.35 + 1.08i)T \)
67 \( 1 + (6.56 - 4.89i)T \)
good2 \( 1 + (0.314 - 0.544i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3.91T + 5T^{2} \)
7 \( 1 + (3.42 - 1.97i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0182 - 0.0316i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.23 + 2.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.19 + 1.26i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 + 1.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.21 - 1.85i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.92 + 3.42i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.21 - 3.58i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.187 - 0.323i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.288 + 0.499i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4.64iT - 43T^{2} \)
47 \( 1 + (-0.127 + 0.0735i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.26T + 53T^{2} \)
59 \( 1 - 2.33iT - 59T^{2} \)
61 \( 1 + (-12.4 - 7.20i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (3.38 - 1.95i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.95 - 8.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.91 - 5.14i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-14.2 - 8.23i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 17.8iT - 89T^{2} \)
97 \( 1 + (1.37 + 0.796i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.89865894486286193502131773849, −11.95084115356760122737394821718, −10.11460836125327677279701022173, −9.320214411255555297286221280402, −8.736924871931357374797144452530, −7.22194153021576829882832966075, −6.56552157275338577088318100854, −5.56364578636807780554112118191, −2.98668596068309882978570932111, −2.40568185350040596735061665053, 1.96056257285753109791412211280, 3.04559190295115874795184564249, 4.89091841759450446686136460201, 6.18951166539172330032421254806, 7.04102826302579797111138821116, 9.064136078157504710158332050529, 9.625293891659553387339015657061, 10.13188491058813725754408870637, 10.86477265866239630718108463253, 12.61331773904597445295863865421

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