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Properties

Label 2-201-67.66-c4-0-13
Degree $2$
Conductor $201$
Sign $0.475 - 0.879i$
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  − 0.951i·2-s − 5.19i·3-s + 15.0·4-s + 27.2i·5-s − 4.94·6-s − 0.640i·7-s − 29.5i·8-s − 27·9-s + 25.9·10-s + 120. i·11-s − 78.4i·12-s + 247. i·13-s − 0.609·14-s + 141.·15-s + 213.·16-s − 172.·17-s + ⋯
L(s)  = 1  − 0.237i·2-s − 0.577i·3-s + 0.943·4-s + 1.09i·5-s − 0.137·6-s − 0.0130i·7-s − 0.462i·8-s − 0.333·9-s + 0.259·10-s + 0.996i·11-s − 0.544i·12-s + 1.46i·13-s − 0.00310·14-s + 0.629·15-s + 0.833·16-s − 0.596·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.475 - 0.879i$
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: \((2,\ 201,\ (\ :2),\ 0.475 - 0.879i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.957710333\)
\(L(\frac12)\) \(\approx\) \(1.957710333\)
\(L(3)\) 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 \( 1 + (-3.94e3 - 2.13e3i)T \)
good2 \( 1 + 0.951iT - 16T^{2} \)
5 \( 1 - 27.2iT - 625T^{2} \)
7 \( 1 + 0.640iT - 2.40e3T^{2} \)
11 \( 1 - 120. iT - 1.46e4T^{2} \)
13 \( 1 - 247. iT - 2.85e4T^{2} \)
17 \( 1 + 172.T + 8.35e4T^{2} \)
19 \( 1 + 265.T + 1.30e5T^{2} \)
23 \( 1 + 339.T + 2.79e5T^{2} \)
29 \( 1 + 988.T + 7.07e5T^{2} \)
31 \( 1 - 1.63e3iT - 9.23e5T^{2} \)
37 \( 1 - 2.47e3T + 1.87e6T^{2} \)
41 \( 1 - 1.32e3iT - 2.82e6T^{2} \)
43 \( 1 + 3.16e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.86e3T + 4.87e6T^{2} \)
53 \( 1 - 5.11e3iT - 7.89e6T^{2} \)
59 \( 1 + 938.T + 1.21e7T^{2} \)
61 \( 1 + 1.00e3iT - 1.38e7T^{2} \)
71 \( 1 - 4.25e3T + 2.54e7T^{2} \)
73 \( 1 + 9.52e3T + 2.83e7T^{2} \)
79 \( 1 + 1.01e4iT - 3.89e7T^{2} \)
83 \( 1 - 1.12e3T + 4.74e7T^{2} \)
89 \( 1 + 829.T + 6.27e7T^{2} \)
97 \( 1 + 1.06e4iT - 8.85e7T^{2} \)
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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

−11.87064504906749848808241535586, −11.04765679531444091391176613089, −10.29228521841996591576710533044, −9.027317596445813494532626916457, −7.42534021881560210387027827249, −6.92670359450541524401041144813, −6.10718793427170577283436493293, −4.14027821456941007369667464446, −2.59387076399362640725992373452, −1.77572790098419929650234762857, 0.64973397331206724765504787517, 2.50213479970678519999327223027, 3.99071578141157207625567444075, 5.44277071152435075256035923059, 6.10702425477143560992343704093, 7.78758684715350388896629264260, 8.460746243879391194983357208491, 9.619021112702079318002947435922, 10.79739464247420036674053875065, 11.41622573370375038353205709626

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