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Properties

Label 1-161-161.6-r1-0-0
Degree $1$
Conductor $161$
Sign $-0.966 + 0.257i$
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  + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (0.142 + 0.989i)6-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)10-s + (−0.654 + 0.755i)11-s + (0.654 − 0.755i)12-s + (0.959 − 0.281i)13-s + (0.841 − 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.142 + 0.989i)17-s + (0.415 − 0.909i)18-s + (0.142 − 0.989i)19-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (0.142 + 0.989i)6-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)10-s + (−0.654 + 0.755i)11-s + (0.654 − 0.755i)12-s + (0.959 − 0.281i)13-s + (0.841 − 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.142 + 0.989i)17-s + (0.415 − 0.909i)18-s + (0.142 − 0.989i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.966 + 0.257i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ -0.966 + 0.257i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003302550850 + 0.02520801831i\)
\(L(\frac12)\) \(\approx\) \(0.003302550850 + 0.02520801831i\)
\(L(1)\) \(\approx\) \(0.4540421884 - 0.1007544017i\)
\(L(1)\) \(\approx\) \(0.4540421884 - 0.1007544017i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + (0.415 + 0.909i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (-0.415 + 0.909i)T \)
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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

−27.78331770654726930525696291578, −27.30232013233811523410444755850, −26.325450138222518837761552097649, −25.15768254977377324276992426747, −24.02706302707112547039717260251, −23.51370799512554927241661175291, −22.55371839042848872332079654197, −21.06979107088926432474239890830, −20.332876842227467839900451398119, −18.832827829849188522465778397571, −18.1100400676429632109321940017, −16.85961318672777459129270814862, −16.15787294274779195085420100788, −15.760980995640050814923304510536, −14.28764987159208951252092423866, −12.94695023389814831344800474278, −11.55605055891434179671794522189, −10.69147219710645935395520467058, −9.442826644892447525958192384693, −8.56505011609572060778753088929, −7.34738175701374482951665671018, −5.875908508300983552243493769776, −5.21470124209367148637332741911, −3.89494024568583695554646520085, −1.17281996945259077238982087157, 0.0153469610684350823508666214, 1.64941620728516535832167022421, 3.016013943870540628787702038792, 4.49561010697919349156958302790, 6.25059418844430197505759958551, 7.35947514288893167692594246977, 8.2080367283306414187993443547, 9.95095481538448514412793896338, 10.84293592205557538734386470969, 11.48089482464603138251997471865, 12.65359478183981654284183521770, 13.459553959668362302730352183663, 15.23235505593884496822919056985, 16.291910788388859445067204415153, 17.60303275133807313715037991441, 18.11198216365272834919561288326, 18.98691466235289817355191657386, 19.86543882687343352683548578188, 21.184690251464288897489601795193, 22.18373580774406792591756888914, 22.9916691483637405424770232046, 23.8363999680244467713957888224, 25.45409147184122635772199477804, 26.12731523481282648743193440250, 27.26857758604590019847177089659

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