↑

Properties

Label 2-201-3.2-c2-0-18
Degree $2$
Conductor $201$
Sign $0.917 - 0.397i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.846i·2-s + (−2.75 + 1.19i)3-s + 3.28·4-s − 3.42i·5-s + (−1.01 − 2.33i)6-s + 3.13·7-s + 6.16i·8-s + (6.15 − 6.56i)9-s + 2.90·10-s − 11.1i·11-s + (−9.03 + 3.91i)12-s + 5.69·13-s + 2.65i·14-s + (4.08 + 9.42i)15-s + 7.90·16-s + 8.41i·17-s + ⋯
L(s)  = 1  + 0.423i·2-s + (−0.917 + 0.397i)3-s + 0.820·4-s − 0.684i·5-s + (−0.168 − 0.388i)6-s + 0.447·7-s + 0.771i·8-s + (0.683 − 0.729i)9-s + 0.290·10-s − 1.01i·11-s + (−0.752 + 0.326i)12-s + 0.437·13-s + 0.189i·14-s + (0.272 + 0.628i)15-s + 0.494·16-s + 0.494i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.917 - 0.397i$
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: \((2,\ 201,\ (\ :1),\ 0.917 - 0.397i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.50016 + 0.311114i\)
\(L(\frac12)\) \(\approx\) \(1.50016 + 0.311114i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.75 - 1.19i)T \)
67 \( 1 + 8.18T \)
good2 \( 1 - 0.846iT - 4T^{2} \)
5 \( 1 + 3.42iT - 25T^{2} \)
7 \( 1 - 3.13T + 49T^{2} \)
11 \( 1 + 11.1iT - 121T^{2} \)
13 \( 1 - 5.69T + 169T^{2} \)
17 \( 1 - 8.41iT - 289T^{2} \)
19 \( 1 - 31.2T + 361T^{2} \)
23 \( 1 - 30.0iT - 529T^{2} \)
29 \( 1 + 13.7iT - 841T^{2} \)
31 \( 1 + 21.6T + 961T^{2} \)
37 \( 1 + 6.33T + 1.36e3T^{2} \)
41 \( 1 + 29.1iT - 1.68e3T^{2} \)
43 \( 1 - 46.3T + 1.84e3T^{2} \)
47 \( 1 + 58.7iT - 2.20e3T^{2} \)
53 \( 1 + 89.7iT - 2.80e3T^{2} \)
59 \( 1 - 27.0iT - 3.48e3T^{2} \)
61 \( 1 + 69.3T + 3.72e3T^{2} \)
71 \( 1 - 86.1iT - 5.04e3T^{2} \)
73 \( 1 + 42.4T + 5.32e3T^{2} \)
79 \( 1 - 13.3T + 6.24e3T^{2} \)
83 \( 1 - 107. iT - 6.88e3T^{2} \)
89 \( 1 + 76.4iT - 7.92e3T^{2} \)
97 \( 1 + 115.T + 9.40e3T^{2} \)
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

−11.93394640208732980688580498688, −11.40466696845254969787990737726, −10.59341341721957766671338377076, −9.308238229355428127286861455711, −8.142360578905303583617492568008, −7.04431726424221258409020773297, −5.77203983708627443611144066506, −5.26430026389982968103840737619, −3.57054245614803397629657800762, −1.25709009541859399714625921823, 1.39328645445516199318123193102, 2.85381027519693773301384668716, 4.68037323561057075798974763896, 6.04141383292418707894918168420, 7.04295260733553591948323774655, 7.63502409894851631058201038144, 9.565914668575858402774220348383, 10.65987007424533783311252238370, 11.10320970140251482766577089377, 12.08112619471886561302706676840

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