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Properties

Label 2-201-67.66-c2-0-15
Degree $2$
Conductor $201$
Sign $-0.908 + 0.416i$
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  − 2.88i·2-s + 1.73i·3-s − 4.31·4-s + 1.18i·5-s + 4.99·6-s − 3.84i·7-s + 0.914i·8-s − 2.99·9-s + 3.42·10-s − 11.0i·11-s − 7.47i·12-s − 19.7i·13-s − 11.0·14-s − 2.05·15-s − 14.6·16-s + 7.89·17-s + ⋯
L(s)  = 1  − 1.44i·2-s + 0.577i·3-s − 1.07·4-s + 0.237i·5-s + 0.832·6-s − 0.549i·7-s + 0.114i·8-s − 0.333·9-s + 0.342·10-s − 1.00i·11-s − 0.623i·12-s − 1.52i·13-s − 0.792·14-s − 0.137·15-s − 0.914·16-s + 0.464·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.416i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.908 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.908 + 0.416i$
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: \((2,\ 201,\ (\ :1),\ -0.908 + 0.416i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.269236 - 1.23296i\)
\(L(\frac12)\) \(\approx\) \(0.269236 - 1.23296i\)
\(L(2)\) 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 \( 1 + (27.9 + 60.9i)T \)
good2 \( 1 + 2.88iT - 4T^{2} \)
5 \( 1 - 1.18iT - 25T^{2} \)
7 \( 1 + 3.84iT - 49T^{2} \)
11 \( 1 + 11.0iT - 121T^{2} \)
13 \( 1 + 19.7iT - 169T^{2} \)
17 \( 1 - 7.89T + 289T^{2} \)
19 \( 1 + 35.4T + 361T^{2} \)
23 \( 1 - 38.7T + 529T^{2} \)
29 \( 1 + 4.14T + 841T^{2} \)
31 \( 1 + 28.6iT - 961T^{2} \)
37 \( 1 + 30.8T + 1.36e3T^{2} \)
41 \( 1 - 64.0iT - 1.68e3T^{2} \)
43 \( 1 - 51.4iT - 1.84e3T^{2} \)
47 \( 1 + 0.461T + 2.20e3T^{2} \)
53 \( 1 + 39.6iT - 2.80e3T^{2} \)
59 \( 1 - 76.0T + 3.48e3T^{2} \)
61 \( 1 + 60.6iT - 3.72e3T^{2} \)
71 \( 1 + 38.7T + 5.04e3T^{2} \)
73 \( 1 - 55.4T + 5.32e3T^{2} \)
79 \( 1 - 25.6iT - 6.24e3T^{2} \)
83 \( 1 - 47.9T + 6.88e3T^{2} \)
89 \( 1 - 71.8T + 7.92e3T^{2} \)
97 \( 1 - 96.0iT - 9.40e3T^{2} \)
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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

−11.44511308697281220710157341491, −10.72004578492016505547222598370, −10.35924619582599005424455312269, −9.150726256756537897981724307642, −8.105620901317127703027885831980, −6.47659440255811485703729322538, −4.96292431487910762921046781190, −3.63886031643367851191937224532, −2.80370997691542474048078156097, −0.73067358547130662479312371983, 2.10415162551012387135791744341, 4.47390550726609733167447448422, 5.51493356526690543170805699723, 6.84643764019730236218822736856, 7.09229309948326959186962952265, 8.733574757898804849044901637639, 8.911248847618971543163091888013, 10.67788211866708454134383050050, 11.98589161101860361142774945636, 12.73233909826419758432338107330

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