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Properties

Label 1-201-201.53-r0-0-0
Degree $1$
Conductor $201$
Sign $0.0143 - 0.999i$
Analytic cond. $0.933440$
Root an. cond. $0.933440$
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  + (−0.654 + 0.755i)2-s + (−0.142 − 0.989i)4-s + (−0.959 + 0.281i)5-s + (0.654 − 0.755i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)10-s + (−0.959 + 0.281i)11-s + (−0.841 + 0.540i)13-s + (0.142 + 0.989i)14-s + (−0.959 + 0.281i)16-s + (0.142 − 0.989i)17-s + (−0.654 − 0.755i)19-s + (0.415 + 0.909i)20-s + (0.415 − 0.909i)22-s + (−0.415 − 0.909i)23-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (−0.142 − 0.989i)4-s + (−0.959 + 0.281i)5-s + (0.654 − 0.755i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)10-s + (−0.959 + 0.281i)11-s + (−0.841 + 0.540i)13-s + (0.142 + 0.989i)14-s + (−0.959 + 0.281i)16-s + (0.142 − 0.989i)17-s + (−0.654 − 0.755i)19-s + (0.415 + 0.909i)20-s + (0.415 − 0.909i)22-s + (−0.415 − 0.909i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0143 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0143 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.0143 - 0.999i$
Analytic conductor: \(0.933440\)
Root analytic conductor: \(0.933440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 201,\ (0:\ ),\ 0.0143 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2260820013 - 0.2228617224i\)
\(L(\frac12)\) \(\approx\) \(0.2260820013 - 0.2228617224i\)
\(L(1)\) \(\approx\) \(0.5196852629 + 0.04258917027i\)
\(L(1)\) \(\approx\) \(0.5196852629 + 0.04258917027i\)

Euler product

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

−27.44595124639957917776357341629, −26.43747972287320956779361315399, −25.408150712451109990024636864707, −24.29052962117661685433256214629, −23.36998029927048332946417451330, −22.08302438091255271037808145183, −21.282063433503795073623804631788, −20.38364120125684742901821786247, −19.4163589167218100777689079066, −18.69086725121857930050590426347, −17.72781528369462217627677591113, −16.6869746072912725195638804059, −15.62768265289603601065509461256, −14.68459831588827493514462978687, −12.916201916920024459510662907685, −12.34400931812930603771135869176, −11.30464944509767718249456590754, −10.479353559154195363545296252395, −9.14706579981821034996463291741, −8.04516914650132775508211320484, −7.66404230483529753950863013126, −5.5604568650674282029314228112, −4.27710287872435185875642802329, −3.04027217090075154337236799055, −1.72071394238566230475267582136, 0.293661168441825734530510058997, 2.2923139398611878772433390976, 4.27988041504486269055887730083, 5.112576441624746661205869117186, 6.87498357513063861327748601711, 7.49354143634446355333809439698, 8.342638683901677103094656604756, 9.697580066718913106052838426333, 10.76364741286709250758509619306, 11.560770682138433814147910181030, 13.15191008569234488945835683652, 14.45031630282490765246289471270, 15.02874552708132060583834939024, 16.16374707921105939358739220646, 16.90367025388401692957014713264, 18.08390618528244318909386804976, 18.77378334850410913176618126130, 19.89976233281387043760440282801, 20.56532750508000753904399971881, 22.19079114926918714868800984328, 23.34088367419307333826813119889, 23.80300353894516879060755755622, 24.6460128309832206861569898529, 26.0447819528734034895811277924, 26.59409106538398915000167083246

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