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Properties

Label 2-201-67.66-c4-0-31
Degree $2$
Conductor $201$
Sign $0.706 + 0.708i$
Analytic cond. $20.7773$
Root an. cond. $4.55821$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.12i·2-s + 5.19i·3-s + 11.4·4-s + 15.4i·5-s + 11.0·6-s − 14.1i·7-s − 58.4i·8-s − 27·9-s + 32.9·10-s − 140. i·11-s + 59.6i·12-s − 150. i·13-s − 30.0·14-s − 80.5·15-s + 59.6·16-s − 16.5·17-s + ⋯
L(s)  = 1  − 0.531i·2-s + 0.577i·3-s + 0.717·4-s + 0.619i·5-s + 0.306·6-s − 0.288i·7-s − 0.912i·8-s − 0.333·9-s + 0.329·10-s − 1.16i·11-s + 0.414i·12-s − 0.889i·13-s − 0.153·14-s − 0.357·15-s + 0.232·16-s − 0.0572·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.708i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.706 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.706 + 0.708i$
Analytic conductor: \(20.7773\)
Root analytic conductor: \(4.55821\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :2),\ 0.706 + 0.708i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.336457348\)
\(L(\frac12)\) \(\approx\) \(2.336457348\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19iT \)
67 \( 1 + (-3.17e3 + 3.16e3i)T \)
good2 \( 1 + 2.12iT - 16T^{2} \)
5 \( 1 - 15.4iT - 625T^{2} \)
7 \( 1 + 14.1iT - 2.40e3T^{2} \)
11 \( 1 + 140. iT - 1.46e4T^{2} \)
13 \( 1 + 150. iT - 2.85e4T^{2} \)
17 \( 1 + 16.5T + 8.35e4T^{2} \)
19 \( 1 - 424.T + 1.30e5T^{2} \)
23 \( 1 - 299.T + 2.79e5T^{2} \)
29 \( 1 - 484.T + 7.07e5T^{2} \)
31 \( 1 - 1.37e3iT - 9.23e5T^{2} \)
37 \( 1 - 130.T + 1.87e6T^{2} \)
41 \( 1 - 262. iT - 2.82e6T^{2} \)
43 \( 1 + 1.70e3iT - 3.41e6T^{2} \)
47 \( 1 + 143.T + 4.87e6T^{2} \)
53 \( 1 + 1.87e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.58e3T + 1.21e7T^{2} \)
61 \( 1 + 825. iT - 1.38e7T^{2} \)
71 \( 1 + 7.30e3T + 2.54e7T^{2} \)
73 \( 1 - 3.02e3T + 2.83e7T^{2} \)
79 \( 1 + 3.78e3iT - 3.89e7T^{2} \)
83 \( 1 + 5.51e3T + 4.74e7T^{2} \)
89 \( 1 + 3.43e3T + 6.27e7T^{2} \)
97 \( 1 - 1.56e4iT - 8.85e7T^{2} \)
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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

−11.38489841539045338290239648199, −10.69584110103174868729039015437, −10.13180390416826847916801692469, −8.802832847267183437084253666795, −7.52091277940976003294473371660, −6.47023086087738148378153779031, −5.27587784793153515414345663050, −3.46978370907479306551778288910, −2.88812160934866511360135741230, −0.917392928394786033260571891280, 1.37658224354923701994236123508, 2.60696311902133449130643790211, 4.63166797398390694708932568590, 5.78050933185902951815495949431, 6.90821457564889822900739678267, 7.58757815670448902857465805541, 8.737501757346261320402774857718, 9.762121646363257887226874747674, 11.21966689844843014175097934869, 11.98132192105626270014243086807

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