Properties

Label 1-161-161.146-r1-0-0
Degree $1$
Conductor $161$
Sign $-0.682 + 0.731i$
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  + (−0.959 − 0.281i)2-s + (0.654 − 0.755i)3-s + (0.841 + 0.540i)4-s + (0.142 − 0.989i)5-s + (−0.841 + 0.540i)6-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (−0.415 + 0.909i)10-s + (−0.959 + 0.281i)11-s + (0.959 − 0.281i)12-s + (−0.415 + 0.909i)13-s + (−0.654 − 0.755i)15-s + (0.415 + 0.909i)16-s + (−0.841 + 0.540i)17-s + (−0.142 + 0.989i)18-s + (−0.841 − 0.540i)19-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.654 − 0.755i)3-s + (0.841 + 0.540i)4-s + (0.142 − 0.989i)5-s + (−0.841 + 0.540i)6-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (−0.415 + 0.909i)10-s + (−0.959 + 0.281i)11-s + (0.959 − 0.281i)12-s + (−0.415 + 0.909i)13-s + (−0.654 − 0.755i)15-s + (0.415 + 0.909i)16-s + (−0.841 + 0.540i)17-s + (−0.142 + 0.989i)18-s + (−0.841 − 0.540i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.682 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.682 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.682 + 0.731i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ -0.682 + 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1468714723 - 0.3379120655i\)
\(L(\frac12)\) \(\approx\) \(-0.1468714723 - 0.3379120655i\)
\(L(1)\) \(\approx\) \(0.5555980145 - 0.3540465035i\)
\(L(1)\) \(\approx\) \(0.5555980145 - 0.3540465035i\)

Euler product

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

−27.76337426245287936544026218675, −26.98010160955277894878425286420, −26.36726886067828038111428593358, −25.535261874388819641840388700652, −24.76342721631151549870708463020, −23.34489268379277489356169482238, −22.17607700013052939104678246445, −21.11961510806938896384289873422, −20.19564527178046699649246388844, −19.22001935391902134489449752230, −18.356666973725971589424005419191, −17.3552182640634753930082655569, −16.07754540454296116624238637189, −15.28678150401066272080042106597, −14.58781741572645263065195030557, −13.33658389817725264609745776909, −11.39396249966626817732661599560, −10.39820193604795815852538002322, −9.93842837267867478696968708404, −8.504789066659357109500411001563, −7.72788819684197758670034079583, −6.43344789579990459499813736355, −5.05142119737175584627632495704, −3.11771691573136980878285094597, −2.292380039296293162604914486053, 0.15615333765681950410900849206, 1.65948409225607641595723533287, 2.58087324571867491945459040957, 4.383574999899325764069858864687, 6.29755812729340345370812311995, 7.42337185490485616416310953094, 8.49906247050975849905417532501, 9.092785632026284096286451926983, 10.336837223600508518063786924092, 11.821231214142220163246748362651, 12.682003490479272240895917339935, 13.51727655456302798903323631057, 15.106861057501025774628967871031, 16.13389811350267711604641377105, 17.35688792944550207697455904713, 17.97482577356779973718415544903, 19.31552852641022229926717558139, 19.71532720393290069192714938342, 20.902955460058886527833725795346, 21.43259332170946635024503520899, 23.524646057355399255164168207529, 24.29444136505619218890550199396, 25.083455939621832977978158016186, 26.06711979769482541711658182862, 26.71254045980168214869197534675

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