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Properties

Label 1-201-201.53-r0-0-0
Degree 11
Conductor 201201
Sign 0.0143−0.999i0.0143 - 0.999i
Analytic cond. 0.9334400.933440
Root an. cond. 0.9334400.933440
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(201s/2 ΓR(s) L(s)=((0.0143−0.999i) Λ‾(1−s)\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}
Λ(s)=(201s/2 ΓR(s) L(s)=((0.0143−0.999i) Λ‾(1−s)\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: 11
Conductor: 201201    =    3⋅673 \cdot 67
Sign: 0.0143−0.999i0.0143 - 0.999i
Analytic conductor: 0.9334400.933440
Root analytic conductor: 0.9334400.933440
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ201(53,⋅)\chi_{201} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 201, (0: ), 0.0143−0.999i)(1,\ 201,\ (0:\ ),\ 0.0143 - 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.2260820013−0.2228617224i0.2260820013 - 0.2228617224i
L(12)L(\frac12) ≈\approx 0.2260820013−0.2228617224i0.2260820013 - 0.2228617224i
L(1)L(1) ≈\approx 0.5196852629+0.04258917027i0.5196852629 + 0.04258917027i
L(1)L(1) ≈\approx 0.5196852629+0.04258917027i0.5196852629 + 0.04258917027i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
67 1 1
good2 1+(−0.654+0.755i)T 1 + (-0.654 + 0.755i)T
5 1+(−0.959+0.281i)T 1 + (-0.959 + 0.281i)T
7 1+(0.654−0.755i)T 1 + (0.654 - 0.755i)T
11 1+(−0.959+0.281i)T 1 + (-0.959 + 0.281i)T
13 1+(−0.841+0.540i)T 1 + (-0.841 + 0.540i)T
17 1+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
19 1+(−0.654−0.755i)T 1 + (-0.654 - 0.755i)T
23 1+(−0.415−0.909i)T 1 + (-0.415 - 0.909i)T
29 1−T 1 - T
31 1+(−0.841−0.540i)T 1 + (-0.841 - 0.540i)T
37 1+T 1 + T
41 1+(−0.142+0.989i)T 1 + (-0.142 + 0.989i)T
43 1+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
47 1+(−0.415−0.909i)T 1 + (-0.415 - 0.909i)T
53 1+(−0.142−0.989i)T 1 + (-0.142 - 0.989i)T
59 1+(−0.841−0.540i)T 1 + (-0.841 - 0.540i)T
61 1+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
71 1+(0.142+0.989i)T 1 + (0.142 + 0.989i)T
73 1+(−0.959−0.281i)T 1 + (-0.959 - 0.281i)T
79 1+(−0.841+0.540i)T 1 + (-0.841 + 0.540i)T
83 1+(0.959−0.281i)T 1 + (0.959 - 0.281i)T
89 1+(−0.415+0.909i)T 1 + (-0.415 + 0.909i)T
97 1−T 1 - T
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   L(s)=∏p (1−αp p−s)−1L(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 ZZ-function along the critical line