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Properties

Label 2-201-67.38-c2-0-9
Degree $2$
Conductor $201$
Sign $0.846 + 0.532i$
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  + (−0.893 − 0.515i)2-s − 1.73i·3-s + (−1.46 − 2.54i)4-s + 8.85i·5-s + (−0.893 + 1.54i)6-s + (7.85 − 4.53i)7-s + 7.15i·8-s − 2.99·9-s + (4.56 − 7.90i)10-s + (3.62 − 2.09i)11-s + (−4.40 + 2.54i)12-s + (13.1 + 7.56i)13-s − 9.35·14-s + 15.3·15-s + (−2.18 + 3.78i)16-s + (−0.813 + 1.40i)17-s + ⋯
L(s)  = 1  + (−0.446 − 0.257i)2-s − 0.577i·3-s + (−0.367 − 0.635i)4-s + 1.77i·5-s + (−0.148 + 0.257i)6-s + (1.12 − 0.647i)7-s + 0.894i·8-s − 0.333·9-s + (0.456 − 0.790i)10-s + (0.329 − 0.190i)11-s + (−0.367 + 0.211i)12-s + (1.00 + 0.582i)13-s − 0.668·14-s + 1.02·15-s + (−0.136 + 0.236i)16-s + (−0.0478 + 0.0828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.846 + 0.532i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.846 + 0.532i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.22712 - 0.354259i\)
\(L(\frac12)\) \(\approx\) \(1.22712 - 0.354259i\)
\(L(2)\) 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 \( 1 + (18.3 + 64.4i)T \)
good2 \( 1 + (0.893 + 0.515i)T + (2 + 3.46i)T^{2} \)
5 \( 1 - 8.85iT - 25T^{2} \)
7 \( 1 + (-7.85 + 4.53i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-3.62 + 2.09i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-13.1 - 7.56i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (0.813 - 1.40i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-15.2 + 26.4i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-12.8 + 22.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-11.6 - 20.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-31.3 + 18.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (4.84 - 8.39i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-29.7 + 17.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 77.4iT - 1.84e3T^{2} \)
47 \( 1 + (7.25 + 12.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 65.4iT - 2.80e3T^{2} \)
59 \( 1 + 40.1T + 3.48e3T^{2} \)
61 \( 1 + (-12.1 - 7.03i)T + (1.86e3 + 3.22e3i)T^{2} \)
71 \( 1 + (8.72 + 15.1i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (35.2 - 61.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-1.93 + 1.11i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-16.3 + 28.2i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 168.T + 7.92e3T^{2} \)
97 \( 1 + (-162. - 93.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.53433939540634134627964202125, −11.09417874179194400149755788347, −10.50339481174831199011692659635, −9.200938479529134856025959166177, −8.070943948270037755311193456389, −6.98675216992875432638608588856, −6.16057123182229729435861579966, −4.53249283239052747203175237890, −2.75997286802947663882266321725, −1.21938051314201155457804749622, 1.22114251009525198936242154224, 3.75219334207374762858032494283, 4.82559833742510083043209016298, 5.69213836134438039859685934338, 7.81837016431826195805904974413, 8.442078145533053580711875858357, 9.027420458866270919037269945173, 9.987391491444859478358356513929, 11.61453308699716894879389896441, 12.18560553915465922817794309672

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