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Properties

Label 1-161-161.33-r0-0-0
Degree 11
Conductor 161161
Sign 0.970−0.242i0.970 - 0.242i
Analytic cond. 0.7476800.747680
Root an. cond. 0.7476800.747680
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  + (0.928 − 0.371i)2-s + (0.995 + 0.0950i)3-s + (0.723 − 0.690i)4-s + (−0.327 + 0.945i)5-s + (0.959 − 0.281i)6-s + (0.415 − 0.909i)8-s + (0.981 + 0.189i)9-s + (0.0475 + 0.998i)10-s + (−0.928 − 0.371i)11-s + (0.786 − 0.618i)12-s + (−0.841 + 0.540i)13-s + (−0.415 + 0.909i)15-s + (0.0475 − 0.998i)16-s + (0.235 − 0.971i)17-s + (0.981 − 0.189i)18-s + (0.235 + 0.971i)19-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)2-s + (0.995 + 0.0950i)3-s + (0.723 − 0.690i)4-s + (−0.327 + 0.945i)5-s + (0.959 − 0.281i)6-s + (0.415 − 0.909i)8-s + (0.981 + 0.189i)9-s + (0.0475 + 0.998i)10-s + (−0.928 − 0.371i)11-s + (0.786 − 0.618i)12-s + (−0.841 + 0.540i)13-s + (−0.415 + 0.909i)15-s + (0.0475 − 0.998i)16-s + (0.235 − 0.971i)17-s + (0.981 − 0.189i)18-s + (0.235 + 0.971i)19-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓR(s) L(s)=((0.970−0.242i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(161s/2 ΓR(s) L(s)=((0.970−0.242i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 161161    =    7⋅237 \cdot 23
Sign: 0.970−0.242i0.970 - 0.242i
Analytic conductor: 0.7476800.747680
Root analytic conductor: 0.7476800.747680
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ161(33,⋅)\chi_{161} (33, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 161, (0: ), 0.970−0.242i)(1,\ 161,\ (0:\ ),\ 0.970 - 0.242i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.318700838−0.2849407876i2.318700838 - 0.2849407876i
L(12)L(\frac12) ≈\approx 2.318700838−0.2849407876i2.318700838 - 0.2849407876i
L(1)L(1) ≈\approx 2.048368748−0.2196474877i2.048368748 - 0.2196474877i
L(1)L(1) ≈\approx 2.048368748−0.2196474877i2.048368748 - 0.2196474877i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
23 1 1
good2 1+(0.928−0.371i)T 1 + (0.928 - 0.371i)T
3 1+(0.995+0.0950i)T 1 + (0.995 + 0.0950i)T
5 1+(−0.327+0.945i)T 1 + (-0.327 + 0.945i)T
11 1+(−0.928−0.371i)T 1 + (-0.928 - 0.371i)T
13 1+(−0.841+0.540i)T 1 + (-0.841 + 0.540i)T
17 1+(0.235−0.971i)T 1 + (0.235 - 0.971i)T
19 1+(0.235+0.971i)T 1 + (0.235 + 0.971i)T
29 1+(−0.959+0.281i)T 1 + (-0.959 + 0.281i)T
31 1+(−0.580−0.814i)T 1 + (-0.580 - 0.814i)T
37 1+(−0.981−0.189i)T 1 + (-0.981 - 0.189i)T
41 1+(0.654+0.755i)T 1 + (0.654 + 0.755i)T
43 1+(−0.415−0.909i)T 1 + (-0.415 - 0.909i)T
47 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
53 1+(0.888−0.458i)T 1 + (0.888 - 0.458i)T
59 1+(−0.0475−0.998i)T 1 + (-0.0475 - 0.998i)T
61 1+(−0.995+0.0950i)T 1 + (-0.995 + 0.0950i)T
67 1+(0.786+0.618i)T 1 + (0.786 + 0.618i)T
71 1+(−0.142+0.989i)T 1 + (-0.142 + 0.989i)T
73 1+(−0.723+0.690i)T 1 + (-0.723 + 0.690i)T
79 1+(0.888+0.458i)T 1 + (0.888 + 0.458i)T
83 1+(−0.654+0.755i)T 1 + (-0.654 + 0.755i)T
89 1+(0.580−0.814i)T 1 + (0.580 - 0.814i)T
97 1+(−0.654−0.755i)T 1 + (-0.654 - 0.755i)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

−27.790458886023183978085509737555, −26.45361447943180413067426148762, −25.768219756380943070178928952279, −24.65592728499161548032552245241, −24.1489708644870706197065157472, −23.22024031547833390906511184466, −21.82030229297743594640837223262, −20.97685856968788052013187314976, −20.1531911078527986769127491758, −19.44944711902787567360711796925, −17.777454725167370595589077263913, −16.60197750720280766554934352415, −15.46954906662778384585487872510, −14.99530579600406060867647943292, −13.638061945016323834096611988947, −12.86118572329763924968791922485, −12.19450356122251157415521499604, −10.502551669500252086666673016463, −9.01290408520415896362085527165, −7.93444488953155210789903085365, −7.23510360777051376590383065704, −5.43789389985252006373425667699, −4.484616240899181277891354203448, −3.27158459294185360903244545581, −1.99210984553606555076089587722, 2.127548554919805833124574570129, 3.028402719386306222545655225386, 4.01977818208712066410973650433, 5.424493625955204940424465709950, 7.01715784680665699694292363922, 7.75402664164098899197362721864, 9.56100537031416893786742538825, 10.45878844710511495537355670155, 11.58449068195500671274492453980, 12.77445166585338777093419393412, 13.92507692185224744652840079598, 14.49806350071164565351080179759, 15.42461744564856523552002579471, 16.33005576377385676676690654581, 18.54043974223110068533885472391, 18.94649557393386163875667176880, 20.09614230764193821790635167740, 20.93579900562493019762787836982, 21.852757035778388132613298289654, 22.74623058670269960569125633227, 23.84852115844265970342232311942, 24.707713677522643732802041659548, 25.803627261016900386771361858874, 26.66833577760346630198847796794, 27.619288536994755187090167827751

Graph of the ZZ-function along the critical line