Properties

Label 2-201-3.2-c2-0-39
Degree $2$
Conductor $201$
Sign $-0.932 + 0.361i$
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  − 2.83i·2-s + (2.79 − 1.08i)3-s − 4.02·4-s − 6.23i·5-s + (−3.07 − 7.92i)6-s + 3.53·7-s + 0.0785i·8-s + (6.64 − 6.07i)9-s − 17.6·10-s + 17.9i·11-s + (−11.2 + 4.37i)12-s + 10.1·13-s − 10.0i·14-s + (−6.77 − 17.4i)15-s − 15.8·16-s + 14.3i·17-s + ⋯
L(s)  = 1  − 1.41i·2-s + (0.932 − 0.361i)3-s − 1.00·4-s − 1.24i·5-s + (−0.512 − 1.32i)6-s + 0.504·7-s + 0.00981i·8-s + (0.738 − 0.674i)9-s − 1.76·10-s + 1.63i·11-s + (−0.938 + 0.364i)12-s + 0.781·13-s − 0.715i·14-s + (−0.451 − 1.16i)15-s − 0.993·16-s + 0.846i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.932 + 0.361i$
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: \((2,\ 201,\ (\ :1),\ -0.932 + 0.361i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.392825 - 2.09740i\)
\(L(\frac12)\) \(\approx\) \(0.392825 - 2.09740i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.79 + 1.08i)T \)
67 \( 1 + 8.18T \)
good2 \( 1 + 2.83iT - 4T^{2} \)
5 \( 1 + 6.23iT - 25T^{2} \)
7 \( 1 - 3.53T + 49T^{2} \)
11 \( 1 - 17.9iT - 121T^{2} \)
13 \( 1 - 10.1T + 169T^{2} \)
17 \( 1 - 14.3iT - 289T^{2} \)
19 \( 1 + 20.3T + 361T^{2} \)
23 \( 1 + 17.7iT - 529T^{2} \)
29 \( 1 - 33.6iT - 841T^{2} \)
31 \( 1 + 5.31T + 961T^{2} \)
37 \( 1 + 4.65T + 1.36e3T^{2} \)
41 \( 1 - 48.2iT - 1.68e3T^{2} \)
43 \( 1 - 68.3T + 1.84e3T^{2} \)
47 \( 1 + 52.1iT - 2.20e3T^{2} \)
53 \( 1 + 54.0iT - 2.80e3T^{2} \)
59 \( 1 - 25.5iT - 3.48e3T^{2} \)
61 \( 1 - 76.4T + 3.72e3T^{2} \)
71 \( 1 + 17.8iT - 5.04e3T^{2} \)
73 \( 1 - 126.T + 5.32e3T^{2} \)
79 \( 1 + 95.7T + 6.24e3T^{2} \)
83 \( 1 + 75.3iT - 6.88e3T^{2} \)
89 \( 1 - 89.3iT - 7.92e3T^{2} \)
97 \( 1 + 119.T + 9.40e3T^{2} \)
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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

−12.10432153445908898172168025737, −10.80424912418343472960522501744, −9.842460224008086077824903929288, −8.904348547923796857182704089052, −8.236070835374003724377705274675, −6.78000186800994341872457005995, −4.70324173500795076949073504397, −3.90827219116620244560077980341, −2.16906838531393154170819455260, −1.31222967809557771318984716174, 2.66030158437476978093337400459, 3.98994243222703325026907739143, 5.60354859018756114552522910669, 6.59052719418838891988131922683, 7.64328761480959744739397153555, 8.362240459863529510971687596566, 9.262428893516389983426070316512, 10.77912040594532316951045097754, 11.26449736577483280651270307284, 13.33885723048472494045722833766

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