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Properties

Label 1-201-201.107-r1-0-0
Degree $1$
Conductor $201$
Sign $0.709 - 0.704i$
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  + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.841 − 0.540i)5-s + (−0.142 + 0.989i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)10-s + (−0.841 − 0.540i)11-s + (0.415 + 0.909i)13-s + (0.959 + 0.281i)14-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + (−0.142 − 0.989i)19-s + (0.654 + 0.755i)20-s + (−0.654 + 0.755i)22-s + (0.654 + 0.755i)23-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.841 − 0.540i)5-s + (−0.142 + 0.989i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)10-s + (−0.841 − 0.540i)11-s + (0.415 + 0.909i)13-s + (0.959 + 0.281i)14-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + (−0.142 − 0.989i)19-s + (0.654 + 0.755i)20-s + (−0.654 + 0.755i)22-s + (0.654 + 0.755i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.709 - 0.704i$
Analytic conductor: \(21.6004\)
Root analytic conductor: \(21.6004\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 201,\ (1:\ ),\ 0.709 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.146247699 - 0.4724760902i\)
\(L(\frac12)\) \(\approx\) \(1.146247699 - 0.4724760902i\)
\(L(1)\) \(\approx\) \(0.8082091995 - 0.3732655514i\)
\(L(1)\) \(\approx\) \(0.8082091995 - 0.3732655514i\)

Euler product

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

−26.65783135394762275626955221493, −25.86476904099438104965386597267, −24.96946964775565187096361668628, −23.63288123698428726024814303745, −23.152247135534811170447869919694, −22.62013512878655759130813198566, −21.11844326223632057193038128885, −20.09411275820296425902411702186, −18.83022229301939483354488957514, −18.12974292459116676287135433485, −16.912555324432205463639902099855, −16.13245872301576382978789307084, −15.110518530283715655972809978670, −14.43676863081987731301125856267, −13.1914565348247579982771239711, −12.37939429445369669136847428818, −10.7237973643220412130378440561, −9.98245058090883535650531143572, −8.222128581487104733225169195105, −7.659556199024587608930365567245, −6.71094024114766304814003515674, −5.39772863040307031805469259814, −4.117808813969726399929157892350, −3.21550359017672807966267793389, −0.62044266161014960683572930918, 0.83518459625097134655474403359, 2.459049809441171939378033500121, 3.57851500082463261216568641220, 4.826216491778506748182979619010, 5.78481496443997430672018049139, 7.72075166586118105697388151180, 8.80399116157203896722261984291, 9.51894839612342809859266740051, 11.13892269573694015038976248710, 11.640994185884680893324283400248, 12.7145029648627178961967999034, 13.49010155458249238711945575988, 14.88483886224965438993566863013, 15.81166560551020960685667124001, 16.92722293434075628475539857487, 18.46124614261334904090918431589, 18.900806721882261665817642438105, 19.82289338460632055699920443855, 21.023026303909956289424846819033, 21.46278454059425829987353007903, 22.69992535220316300313935682770, 23.61416726254760323013492137345, 24.29123844349032713793146469329, 25.780796881346326522292441935844, 26.76520049557103919897558057246

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