↑

Properties

Label 1-161-161.34-r0-0-0
Degree $1$
Conductor $161$
Sign $0.206 - 0.978i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.415 − 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (0.654 − 0.755i)6-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.415 − 0.909i)11-s + (−0.415 − 0.909i)12-s + (0.142 + 0.989i)13-s + (0.959 − 0.281i)15-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + (0.841 − 0.540i)18-s + (−0.654 − 0.755i)19-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (0.654 − 0.755i)6-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.415 − 0.909i)11-s + (−0.415 − 0.909i)12-s + (0.142 + 0.989i)13-s + (0.959 − 0.281i)15-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + (0.841 − 0.540i)18-s + (−0.654 − 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.206 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.444941058 - 1.171402142i\)
\(L(\frac12)\) \(\approx\) \(1.444941058 - 1.171402142i\)
\(L(1)\) \(\approx\) \(1.467359430 - 0.7929838556i\)
\(L(1)\) \(\approx\) \(1.467359430 - 0.7929838556i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−27.63975560917093674326015474924, −26.57230332695397124768472802774, −25.81987766686153410976810085680, −25.152316177734931470421071619393, −24.490161588281787387919691696898, −23.118564119714126079816573551024, −22.38433017443234951748755735, −21.11376830022297252942634902166, −20.47168107397236563714477458022, −18.85057243580732483900210114185, −18.01352216084404727100737328486, −17.24787258276135568457216578518, −15.59030232098135170797979693359, −15.03038971434048357118694464513, −13.97644141790466874977243799960, −13.276825674546755007352674628651, −12.35153826790879696682462022883, −10.30173805891585246828239524022, −9.333066899504887074608141760421, −8.13494920130819285048536453650, −7.18344270248980877354144296775, −6.18806288990405561159299695316, −4.8101796062743985061962946485, −3.33643280392528595279302352388, −2.199757600934358464222329909613, 1.61714311619683115454450947129, 2.62502211751975594174894025058, 3.97949696465122630676246299156, 5.02728392066149143580876291181, 6.41318920375873996716273899814, 8.524148456339185740254723302956, 9.07653995098762723585581081558, 10.18830046636760033997430276382, 11.16992857060393051589972404500, 12.75998136213942836137690045578, 13.46238399333787825194504759350, 14.17098580264904780029320516119, 15.311744917840645450215282409857, 16.60338655376234296327445960595, 17.98847023840254214167078588814, 19.1044184293308954762688990238, 19.796923107994251714537994331997, 21.0630390036866804852863720129, 21.30888946395365110978351137218, 22.23288839830537326075615058764, 23.975676638916750918858914136029, 24.36387332662696243276787861860, 25.87069733911384633949568994696, 26.538384201708239758461021773735, 27.78733062474151367633228858706

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