↑

Properties

Label 2-201-67.66-c2-0-16
Degree $2$
Conductor $201$
Sign $-0.680 + 0.732i$
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  − 1.70i·2-s − 1.73i·3-s + 1.09·4-s + 1.67i·5-s − 2.95·6-s + 0.231i·7-s − 8.68i·8-s − 2.99·9-s + 2.85·10-s − 17.0i·11-s − 1.90i·12-s − 6.17i·13-s + 0.394·14-s + 2.89·15-s − 10.4·16-s + 15.1·17-s + ⋯
L(s)  = 1  − 0.851i·2-s − 0.577i·3-s + 0.274·4-s + 0.334i·5-s − 0.491·6-s + 0.0330i·7-s − 1.08i·8-s − 0.333·9-s + 0.285·10-s − 1.55i·11-s − 0.158i·12-s − 0.475i·13-s + 0.0281·14-s + 0.193·15-s − 0.650·16-s + 0.889·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.680 + 0.732i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.680 + 0.732i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.654309 - 1.50089i\)
\(L(\frac12)\) \(\approx\) \(0.654309 - 1.50089i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
67 \( 1 + (-49.0 - 45.6i)T \)
good2 \( 1 + 1.70iT - 4T^{2} \)
5 \( 1 - 1.67iT - 25T^{2} \)
7 \( 1 - 0.231iT - 49T^{2} \)
11 \( 1 + 17.0iT - 121T^{2} \)
13 \( 1 + 6.17iT - 169T^{2} \)
17 \( 1 - 15.1T + 289T^{2} \)
19 \( 1 + 6.48T + 361T^{2} \)
23 \( 1 + 37.1T + 529T^{2} \)
29 \( 1 - 24.7T + 841T^{2} \)
31 \( 1 - 48.8iT - 961T^{2} \)
37 \( 1 + 20.6T + 1.36e3T^{2} \)
41 \( 1 + 75.9iT - 1.68e3T^{2} \)
43 \( 1 - 60.0iT - 1.84e3T^{2} \)
47 \( 1 - 71.6T + 2.20e3T^{2} \)
53 \( 1 - 88.0iT - 2.80e3T^{2} \)
59 \( 1 + 7.13T + 3.48e3T^{2} \)
61 \( 1 + 41.0iT - 3.72e3T^{2} \)
71 \( 1 + 0.114T + 5.04e3T^{2} \)
73 \( 1 - 58.1T + 5.32e3T^{2} \)
79 \( 1 + 24.2iT - 6.24e3T^{2} \)
83 \( 1 + 3.41T + 6.88e3T^{2} \)
89 \( 1 - 147.T + 7.92e3T^{2} \)
97 \( 1 - 50.6iT - 9.40e3T^{2} \)
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

−12.03055551714879846194971455343, −10.78222543895687934268246604635, −10.42029913439736288373291204085, −8.884428895212389284346455868893, −7.82107851237363993891193163831, −6.63585522596200241456623782868, −5.69116197976866619867642976065, −3.61521199329934502848808837062, −2.63688298846085149103309797286, −0.972994011982118479595565673676, 2.20423514818176257595938542348, 4.18658691834079124057393575947, 5.24943738416078013045712664316, 6.40914394190037991209977847012, 7.46968534079392438035670889904, 8.401750868054159290079758418770, 9.627053273845613617791672949339, 10.43863460607036687718619747398, 11.74336772775278891652044038758, 12.39864168339165510168539074515

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