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Properties

Label 2-201-201.65-c2-0-6
Degree $2$
Conductor $201$
Sign $-0.720 - 0.693i$
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  + (2.87 + 0.845i)3-s + (−3.98 − 0.380i)4-s + (−11.9 + 6.15i)7-s + (7.57 + 4.86i)9-s + (−11.1 − 4.45i)12-s + (−20.3 + 16.0i)13-s + (15.7 + 3.02i)16-s + (15.2 + 7.85i)19-s + (−39.5 + 7.62i)21-s + (−3.55 − 24.7i)25-s + (17.6 + 20.4i)27-s + (49.8 − 19.9i)28-s + (−42.7 − 33.5i)31-s + (−28.2 − 22.2i)36-s + (36.1 + 62.5i)37-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)3-s + (−0.995 − 0.0950i)4-s + (−1.70 + 0.879i)7-s + (0.841 + 0.540i)9-s + (−0.928 − 0.371i)12-s + (−1.56 + 1.23i)13-s + (0.981 + 0.189i)16-s + (0.801 + 0.413i)19-s + (−1.88 + 0.363i)21-s + (−0.142 − 0.989i)25-s + (0.654 + 0.755i)27-s + (1.78 − 0.713i)28-s + (−1.37 − 1.08i)31-s + (−0.786 − 0.618i)36-s + (0.976 + 1.69i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.720 - 0.693i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.720 - 0.693i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.310665 + 0.771402i\)
\(L(\frac12)\) \(\approx\) \(0.310665 + 0.771402i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.87 - 0.845i)T \)
67 \( 1 + (-46.3 - 48.3i)T \)
good2 \( 1 + (3.98 + 0.380i)T^{2} \)
5 \( 1 + (3.55 + 24.7i)T^{2} \)
7 \( 1 + (11.9 - 6.15i)T + (28.4 - 39.9i)T^{2} \)
11 \( 1 + (95.1 - 74.7i)T^{2} \)
13 \( 1 + (20.3 - 16.0i)T + (39.8 - 164. i)T^{2} \)
17 \( 1 + (-283. + 54.6i)T^{2} \)
19 \( 1 + (-15.2 - 7.85i)T + (209. + 294. i)T^{2} \)
23 \( 1 + (470. - 242. i)T^{2} \)
29 \( 1 + (420.5 + 728. i)T^{2} \)
31 \( 1 + (42.7 + 33.5i)T + (226. + 933. i)T^{2} \)
37 \( 1 + (-36.1 - 62.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (549. - 1.58e3i)T^{2} \)
43 \( 1 + (21.3 - 46.6i)T + (-1.21e3 - 1.39e3i)T^{2} \)
47 \( 1 + (-105. + 2.20e3i)T^{2} \)
53 \( 1 + (1.83e3 - 2.12e3i)T^{2} \)
59 \( 1 + (3.33e3 + 980. i)T^{2} \)
61 \( 1 + (-4.37 + 12.6i)T + (-2.92e3 - 2.30e3i)T^{2} \)
71 \( 1 + (-4.94e3 - 954. i)T^{2} \)
73 \( 1 + (-23.3 + 67.4i)T + (-4.18e3 - 3.29e3i)T^{2} \)
79 \( 1 + (-63.8 - 25.5i)T + (4.51e3 + 4.30e3i)T^{2} \)
83 \( 1 + (-6.39e3 - 2.56e3i)T^{2} \)
89 \( 1 + (-6.66e3 + 4.28e3i)T^{2} \)
97 \( 1 + (-94.0 - 162. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.77447919900449128098354286147, −11.94240648211408379430067086388, −9.849692800439367260457834492007, −9.707356241881584981671321297708, −8.943970845464163457923404245460, −7.72243293676710643381024858288, −6.40021077347205743522725524118, −4.95183570491898739467545831134, −3.73906885534001711407125271592, −2.52195223964227485483762431286, 0.41977627276994826727249569870, 3.00539689694892738745312780468, 3.81213990107237855117957911633, 5.36139667268948043259329216077, 7.09120556526202566369983515558, 7.65133867278636363517918279023, 9.131281155265438948783577467719, 9.650608340327097903531600466052, 10.41645249683701314653972596192, 12.54536569210730513104179168407

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