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Properties

Label 1-201-201.92-r1-0-0
Degree $1$
Conductor $201$
Sign $-0.899 + 0.436i$
Analytic cond. $21.6004$
Root an. cond. $21.6004$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (−0.415 + 0.909i)5-s + (−0.959 + 0.281i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)10-s + (−0.415 + 0.909i)11-s + (−0.654 − 0.755i)13-s + (−0.841 + 0.540i)14-s + (0.415 − 0.909i)16-s + (−0.841 − 0.540i)17-s + (−0.959 − 0.281i)19-s + (0.142 + 0.989i)20-s + (−0.142 + 0.989i)22-s + (0.142 + 0.989i)23-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (−0.415 + 0.909i)5-s + (−0.959 + 0.281i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)10-s + (−0.415 + 0.909i)11-s + (−0.654 − 0.755i)13-s + (−0.841 + 0.540i)14-s + (0.415 − 0.909i)16-s + (−0.841 − 0.540i)17-s + (−0.959 − 0.281i)19-s + (0.142 + 0.989i)20-s + (−0.142 + 0.989i)22-s + (0.142 + 0.989i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.899 + 0.436i$
Analytic conductor: \(21.6004\)
Root analytic conductor: \(21.6004\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 201,\ (1:\ ),\ -0.899 + 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1183585153 + 0.5153339887i\)
\(L(\frac12)\) \(\approx\) \(0.1183585153 + 0.5153339887i\)
\(L(1)\) \(\approx\) \(1.160504875 + 0.04353973050i\)
\(L(1)\) \(\approx\) \(1.160504875 + 0.04353973050i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (-0.654 - 0.755i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (0.142 + 0.989i)T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.415 + 0.909i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (-0.415 + 0.909i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.14614778212576967393655486956, −25.066352941504272706068738442639, −24.04403865403764041449825238949, −23.68517987305359287573394385003, −22.459047875885352710515884252620, −21.665033210486207861559838426279, −20.64351731540286800349452028079, −19.7672724226044623230011472445, −18.903951651514613179953952936323, −16.87392795228601685838614062792, −16.634911629682236597700805356011, −15.61608608031323391942239728977, −14.59148185333890462518521341327, −13.216675202384850293444790106287, −12.89219232708298414408797901040, −11.731987555874438156244118680754, −10.65604972197959520603189443268, −9.09229328707516925857540051347, −8.02970498965513995222027825998, −6.78380885692358791963768409403, −5.80373660111734751711319046741, −4.508558966542256151753042052528, −3.68903341644657277724279021792, −2.20443031162873934327025739222, −0.11536672674949515999047363962, 2.2810701901355412292737219779, 3.093893962117183101077981018574, 4.30961708366834686447853502850, 5.61507314133465341905460261523, 6.79355810110992081171221887942, 7.4971497599152793124850954948, 9.501943105560398402688115416148, 10.456062127967850655664930251015, 11.39118815594096772200768038807, 12.567664825110158894748009829189, 13.18498433010740190326342164771, 14.55927366870673904550790538402, 15.317598883110153157433459136071, 15.92813419387390586697957798977, 17.51910603577829808414882049916, 18.746908027818491881849140538734, 19.61880274837620039764072616703, 20.31919394035279997361302573425, 21.72515999147870080937021508240, 22.38692371926519776006399755717, 23.041472831200436335953875007695, 23.90921934318080107576661327883, 25.299745989088642590277915652611, 25.75433381810070178733587062365, 27.08728622167784397225093660152

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