Properties

Label 2-201-67.66-c4-0-36
Degree $2$
Conductor $201$
Sign $-0.721 + 0.692i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.53i·2-s + 5.19i·3-s + 9.56·4-s − 8.16i·5-s + 13.1·6-s − 69.7i·7-s − 64.8i·8-s − 27·9-s − 20.6·10-s + 42.4i·11-s + 49.7i·12-s − 8.13i·13-s − 176.·14-s + 42.4·15-s − 11.3·16-s − 51.4·17-s + ⋯
L(s)  = 1  − 0.634i·2-s + 0.577i·3-s + 0.598·4-s − 0.326i·5-s + 0.366·6-s − 1.42i·7-s − 1.01i·8-s − 0.333·9-s − 0.206·10-s + 0.351i·11-s + 0.345i·12-s − 0.0481i·13-s − 0.901·14-s + 0.188·15-s − 0.0443·16-s − 0.178·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.721 + 0.692i$
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: \((2,\ 201,\ (\ :2),\ -0.721 + 0.692i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.680306190\)
\(L(\frac12)\) \(\approx\) \(1.680306190\)
\(L(3)\) 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 \( 1 + (-3.10e3 - 3.24e3i)T \)
good2 \( 1 + 2.53iT - 16T^{2} \)
5 \( 1 + 8.16iT - 625T^{2} \)
7 \( 1 + 69.7iT - 2.40e3T^{2} \)
11 \( 1 - 42.4iT - 1.46e4T^{2} \)
13 \( 1 + 8.13iT - 2.85e4T^{2} \)
17 \( 1 + 51.4T + 8.35e4T^{2} \)
19 \( 1 + 552.T + 1.30e5T^{2} \)
23 \( 1 + 661.T + 2.79e5T^{2} \)
29 \( 1 - 952.T + 7.07e5T^{2} \)
31 \( 1 + 1.57e3iT - 9.23e5T^{2} \)
37 \( 1 + 824.T + 1.87e6T^{2} \)
41 \( 1 + 1.04e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.48e3iT - 3.41e6T^{2} \)
47 \( 1 + 3.33e3T + 4.87e6T^{2} \)
53 \( 1 - 5.32e3iT - 7.89e6T^{2} \)
59 \( 1 - 6.33e3T + 1.21e7T^{2} \)
61 \( 1 + 5.49e3iT - 1.38e7T^{2} \)
71 \( 1 + 238.T + 2.54e7T^{2} \)
73 \( 1 + 3.34e3T + 2.83e7T^{2} \)
79 \( 1 + 1.44e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.34e3T + 4.74e7T^{2} \)
89 \( 1 - 4.41e3T + 6.27e7T^{2} \)
97 \( 1 + 4.82e3iT - 8.85e7T^{2} \)
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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

−11.20879420807703523780271757369, −10.41846304635334103281997672038, −9.928490033214189843672178242866, −8.485447233790956744595195321779, −7.26647500479463891169947437898, −6.26453307844078047193562500144, −4.50855565367845268462606925252, −3.72852890685375917029213675777, −2.13217912922474073542220484921, −0.54698395629168946752256277385, 1.92459796854332185249175827260, 2.91875713911829372941252313395, 5.10872881070543422715145787787, 6.26931169099551702588011979290, 6.70960999252055667945098101653, 8.239523481351740287738001207233, 8.628382464266172264032408618828, 10.30161152351251690408601023454, 11.38463318209321177250126500151, 12.10599313490946436046585234897

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