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Properties

Label 1-161-161.30-r1-0-0
Degree $1$
Conductor $161$
Sign $-0.761 - 0.647i$
Analytic cond. $17.3018$
Root an. cond. $17.3018$
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  + (−0.786 + 0.618i)2-s + (0.580 + 0.814i)3-s + (0.235 − 0.971i)4-s + (−0.981 − 0.189i)5-s + (−0.959 − 0.281i)6-s + (0.415 + 0.909i)8-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)10-s + (0.786 + 0.618i)11-s + (0.928 − 0.371i)12-s + (0.841 + 0.540i)13-s + (−0.415 − 0.909i)15-s + (−0.888 − 0.458i)16-s + (−0.723 + 0.690i)17-s + (−0.327 − 0.945i)18-s + (−0.723 − 0.690i)19-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (0.580 + 0.814i)3-s + (0.235 − 0.971i)4-s + (−0.981 − 0.189i)5-s + (−0.959 − 0.281i)6-s + (0.415 + 0.909i)8-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)10-s + (0.786 + 0.618i)11-s + (0.928 − 0.371i)12-s + (0.841 + 0.540i)13-s + (−0.415 − 0.909i)15-s + (−0.888 − 0.458i)16-s + (−0.723 + 0.690i)17-s + (−0.327 − 0.945i)18-s + (−0.723 − 0.690i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.761 - 0.647i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ -0.761 - 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1638682450 + 0.4456141239i\)
\(L(\frac12)\) \(\approx\) \(-0.1638682450 + 0.4456141239i\)
\(L(1)\) \(\approx\) \(0.5209557890 + 0.3894648249i\)
\(L(1)\) \(\approx\) \(0.5209557890 + 0.3894648249i\)

Euler product

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

−27.13132049235944302740614607889, −26.0614200195121534832282988456, −25.20668210475663122894630664845, −24.23698504203541630974287292366, −23.08180530678951288977995569338, −21.99389515943662103815746323241, −20.44063229873374630304408856009, −20.093044313071239910732967831694, −18.86344936417285932234997113585, −18.58755392919404079359137264062, −17.28022359011270645907384036342, −16.12233559667690700626299416599, −14.96750628804425329618237129945, −13.61943530698331361400862855811, −12.57934762287396932601955355087, −11.632126772781606381231867078495, −10.79921408459363614982986975957, −9.10952342911548455962916943260, −8.40583556646356338012926336080, −7.45262562800983346069340418296, −6.41080981003168171988332641166, −3.90826292925475582051293244051, −3.09788725817194612245075191045, −1.5547302337575549757478420718, −0.213758078029453574263687948602, 1.85055787992179845002421033547, 3.8096411496495913399819047013, 4.743420013948561463508664088663, 6.42778493166778910715499189680, 7.64370972587984011584447259473, 8.715617291216065731990906897659, 9.28987261387842616484964317033, 10.7182538465666544712188028047, 11.46714019807679241118525121038, 13.28037886282747722383937044342, 14.78241344158976212748015889430, 15.14400531361443813265684195581, 16.24428197191726335589404719278, 16.91781130449127180910636325725, 18.30491135718910082274721140435, 19.62068604092014141004069164200, 19.82687391927811500962662374316, 21.06530088552720095843368017632, 22.442289876647326055402805841080, 23.46982959024612206647613744603, 24.42662248172613435383883049420, 25.53130284110921353193199833895, 26.27119350013503773346704360528, 27.07628779316268966706363022581, 28.07383101589073923243636784872

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