Properties

Label 2-201-67.66-c4-0-7
Degree $2$
Conductor $201$
Sign $0.292 + 0.956i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6.38i·2-s + 5.19i·3-s − 24.8·4-s + 26.0i·5-s − 33.1·6-s + 27.5i·7-s − 56.2i·8-s − 27·9-s − 166.·10-s + 129. i·11-s − 128. i·12-s + 3.44i·13-s − 175.·14-s − 135.·15-s − 37.6·16-s − 69.5·17-s + ⋯
L(s)  = 1  + 1.59i·2-s + 0.577i·3-s − 1.55·4-s + 1.04i·5-s − 0.921·6-s + 0.562i·7-s − 0.878i·8-s − 0.333·9-s − 1.66·10-s + 1.07i·11-s − 0.894i·12-s + 0.0203i·13-s − 0.897·14-s − 0.602·15-s − 0.147·16-s − 0.240·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.292 + 0.956i$
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: \((2,\ 201,\ (\ :2),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.336791574\)
\(L(\frac12)\) \(\approx\) \(1.336791574\)
\(L(3)\) 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 \( 1 + (-4.29e3 + 1.31e3i)T \)
good2 \( 1 - 6.38iT - 16T^{2} \)
5 \( 1 - 26.0iT - 625T^{2} \)
7 \( 1 - 27.5iT - 2.40e3T^{2} \)
11 \( 1 - 129. iT - 1.46e4T^{2} \)
13 \( 1 - 3.44iT - 2.85e4T^{2} \)
17 \( 1 + 69.5T + 8.35e4T^{2} \)
19 \( 1 + 102.T + 1.30e5T^{2} \)
23 \( 1 - 618.T + 2.79e5T^{2} \)
29 \( 1 - 836.T + 7.07e5T^{2} \)
31 \( 1 + 348. iT - 9.23e5T^{2} \)
37 \( 1 + 758.T + 1.87e6T^{2} \)
41 \( 1 + 72.0iT - 2.82e6T^{2} \)
43 \( 1 - 2.05e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.50e3T + 4.87e6T^{2} \)
53 \( 1 + 1.05e3iT - 7.89e6T^{2} \)
59 \( 1 - 2.50e3T + 1.21e7T^{2} \)
61 \( 1 - 964. iT - 1.38e7T^{2} \)
71 \( 1 - 2.05e3T + 2.54e7T^{2} \)
73 \( 1 + 8.21e3T + 2.83e7T^{2} \)
79 \( 1 - 556. iT - 3.89e7T^{2} \)
83 \( 1 + 8.22e3T + 4.74e7T^{2} \)
89 \( 1 + 1.11e4T + 6.27e7T^{2} \)
97 \( 1 - 7.09e3iT - 8.85e7T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.78996034459297516108078621292, −11.46507788721132411080607034571, −10.38934109477973327017804666318, −9.348957749049725673112631577807, −8.416081969007977726542910444832, −7.21007988092418364744118493685, −6.56200972140917662603673182210, −5.40548758812574565965027462953, −4.36810384202986423606750107074, −2.65817853528457885239713078242, 0.52343853832600102737831564269, 1.29333945151065541072472136816, 2.81936757367333505120445780913, 4.08731757488807923121562044848, 5.29017769444643498803610113591, 6.89830284644408440843801241339, 8.478662379682621320896936208203, 9.003649188841562105700721731855, 10.33216609073046035551759524024, 11.09799962926121117265665680947

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