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Properties

Label 2-201-3.2-c2-0-22
Degree $2$
Conductor $201$
Sign $0.935 - 0.354i$
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  + 3.36i·2-s + (−2.80 + 1.06i)3-s − 7.34·4-s − 7.21i·5-s + (−3.58 − 9.44i)6-s − 0.105·7-s − 11.2i·8-s + (6.73 − 5.96i)9-s + 24.2·10-s + 4.33i·11-s + (20.5 − 7.80i)12-s + 6.10·13-s − 0.353i·14-s + (7.66 + 20.2i)15-s + 8.54·16-s − 25.4i·17-s + ⋯
L(s)  = 1  + 1.68i·2-s + (−0.935 + 0.354i)3-s − 1.83·4-s − 1.44i·5-s + (−0.596 − 1.57i)6-s − 0.0150·7-s − 1.40i·8-s + (0.748 − 0.662i)9-s + 2.42·10-s + 0.394i·11-s + (1.71 − 0.650i)12-s + 0.469·13-s − 0.0252i·14-s + (0.511 + 1.34i)15-s + 0.533·16-s − 1.49i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.935 - 0.354i$
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: \((2,\ 201,\ (\ :1),\ 0.935 - 0.354i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.786900 + 0.144144i\)
\(L(\frac12)\) \(\approx\) \(0.786900 + 0.144144i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.80 - 1.06i)T \)
67 \( 1 + 8.18T \)
good2 \( 1 - 3.36iT - 4T^{2} \)
5 \( 1 + 7.21iT - 25T^{2} \)
7 \( 1 + 0.105T + 49T^{2} \)
11 \( 1 - 4.33iT - 121T^{2} \)
13 \( 1 - 6.10T + 169T^{2} \)
17 \( 1 + 25.4iT - 289T^{2} \)
19 \( 1 + 1.67T + 361T^{2} \)
23 \( 1 + 29.4iT - 529T^{2} \)
29 \( 1 + 8.15iT - 841T^{2} \)
31 \( 1 - 45.3T + 961T^{2} \)
37 \( 1 - 19.4T + 1.36e3T^{2} \)
41 \( 1 + 52.9iT - 1.68e3T^{2} \)
43 \( 1 + 21.1T + 1.84e3T^{2} \)
47 \( 1 - 67.4iT - 2.20e3T^{2} \)
53 \( 1 + 38.7iT - 2.80e3T^{2} \)
59 \( 1 + 58.1iT - 3.48e3T^{2} \)
61 \( 1 - 88.8T + 3.72e3T^{2} \)
71 \( 1 + 123. iT - 5.04e3T^{2} \)
73 \( 1 + 114.T + 5.32e3T^{2} \)
79 \( 1 + 137.T + 6.24e3T^{2} \)
83 \( 1 - 1.25iT - 6.88e3T^{2} \)
89 \( 1 - 64.0iT - 7.92e3T^{2} \)
97 \( 1 + 104.T + 9.40e3T^{2} \)
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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

−12.48340697428967861458817658479, −11.47896433561298629267144834609, −9.905443692453792350698186859458, −9.065213890867439670782740631474, −8.166756177293052865016425737718, −6.94514636685554626845231471537, −5.98567962018525533636589320988, −4.92881390115497781862000304917, −4.47563703370903760508783096900, −0.56287584036881182467757188759, 1.49040432968625748442524558696, 2.97079628163291901403165384090, 4.13119064359600623779091363196, 5.82771722690099929031102387056, 6.85388536919104412593857168943, 8.301547415303684928286954452805, 9.977819506638499395488717330084, 10.43893453749152961641004005895, 11.34424625359749945017063052099, 11.68176206938688715564946013061

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