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Properties

Label 1-161-161.33-r0-0-0
Degree $1$
Conductor $161$
Sign $0.970 - 0.242i$
Analytic cond. $0.747680$
Root an. cond. $0.747680$
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  + (0.928 − 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

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

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.318700838 - 0.2849407876i\)
\(L(\frac12)\) \(\approx\) \(2.318700838 - 0.2849407876i\)
\(L(1)\) \(\approx\) \(2.048368748 - 0.2196474877i\)
\(L(1)\) \(\approx\) \(2.048368748 - 0.2196474877i\)

Euler product

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

−27.790458886023183978085509737555, −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 $Z$-function along the critical line