↑

Properties

Label 1-167-167.39-r1-0-0
Degree $1$
Conductor $167$
Sign $-0.138 + 0.990i$
Analytic cond. $17.9466$
Root an. cond. $17.9466$
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  + (−0.982 + 0.188i)2-s + (−0.942 − 0.334i)3-s + (0.929 − 0.369i)4-s + (0.387 + 0.922i)5-s + (0.988 + 0.150i)6-s + (0.974 + 0.225i)7-s + (−0.843 + 0.537i)8-s + (0.776 + 0.629i)9-s + (−0.553 − 0.832i)10-s + (0.132 + 0.991i)11-s + (−0.999 + 0.0378i)12-s + (0.0944 + 0.995i)13-s + (−0.999 − 0.0378i)14-s + (−0.0567 − 0.998i)15-s + (0.726 − 0.686i)16-s + (0.800 − 0.599i)17-s + ⋯
L(s)  = 1  + (−0.982 + 0.188i)2-s + (−0.942 − 0.334i)3-s + (0.929 − 0.369i)4-s + (0.387 + 0.922i)5-s + (0.988 + 0.150i)6-s + (0.974 + 0.225i)7-s + (−0.843 + 0.537i)8-s + (0.776 + 0.629i)9-s + (−0.553 − 0.832i)10-s + (0.132 + 0.991i)11-s + (−0.999 + 0.0378i)12-s + (0.0944 + 0.995i)13-s + (−0.999 − 0.0378i)14-s + (−0.0567 − 0.998i)15-s + (0.726 − 0.686i)16-s + (0.800 − 0.599i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $-0.138 + 0.990i$
Analytic conductor: \(17.9466\)
Root analytic conductor: \(17.9466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (1:\ ),\ -0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6311303543 + 0.7252096675i\)
\(L(\frac12)\) \(\approx\) \(0.6311303543 + 0.7252096675i\)
\(L(1)\) \(\approx\) \(0.6432570951 + 0.2270666443i\)
\(L(1)\) \(\approx\) \(0.6432570951 + 0.2270666443i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.982 + 0.188i)T \)
3 \( 1 + (-0.942 - 0.334i)T \)
5 \( 1 + (0.387 + 0.922i)T \)
7 \( 1 + (0.974 + 0.225i)T \)
11 \( 1 + (0.132 + 0.991i)T \)
13 \( 1 + (0.0944 + 0.995i)T \)
17 \( 1 + (0.800 - 0.599i)T \)
19 \( 1 + (0.489 - 0.872i)T \)
23 \( 1 + (-0.997 + 0.0756i)T \)
29 \( 1 + (0.997 + 0.0756i)T \)
31 \( 1 + (0.351 - 0.936i)T \)
37 \( 1 + (-0.776 + 0.629i)T \)
41 \( 1 + (0.752 + 0.658i)T \)
43 \( 1 + (0.0189 + 0.999i)T \)
47 \( 1 + (-0.584 - 0.811i)T \)
53 \( 1 + (-0.614 + 0.788i)T \)
59 \( 1 + (0.800 + 0.599i)T \)
61 \( 1 + (-0.455 + 0.890i)T \)
67 \( 1 + (0.387 - 0.922i)T \)
71 \( 1 + (-0.822 + 0.569i)T \)
73 \( 1 + (-0.726 - 0.686i)T \)
79 \( 1 + (0.644 + 0.764i)T \)
83 \( 1 + (0.982 + 0.188i)T \)
89 \( 1 + (0.0567 - 0.998i)T \)
97 \( 1 + (0.351 + 0.936i)T \)
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

−27.49781017521214010958678368730, −26.543611599288346476351863950, −25.10520140300844174749751734807, −24.37221502589097690508875925987, −23.52253183795753278576181799550, −21.948552945989533186300255636492, −21.07148816636202703100779617408, −20.5271621208069866812050606511, −19.171090184407502221408593316094, −17.8390127234096452868803610687, −17.47845234640594521816323760840, −16.423640358452606640249771471471, −15.814562171638939389664344235185, −14.21603428708654060570239445604, −12.55492365793865112077196216099, −11.85386068433675435083386135716, −10.66900968382656977045057225034, −10.016232143230350003996426614843, −8.61119327064176732776423050010, −7.82333820282572672688022504891, −6.10310662958322434734207486075, −5.29302727351295661593115621576, −3.68247995845739183907937991556, −1.52907323177524679351225750609, −0.62518754814201987194865597770, 1.34497517724868808079892335675, 2.391288804719018587239255766351, 4.783884901021925796378438943003, 6.07018058100363449116189959980, 7.013200987437338992654148865537, 7.81249429627079059984180888106, 9.4938838732411176476110269711, 10.35136531126150934165771482400, 11.49846839946935555015349250022, 11.92549484865176494499330338586, 13.86979127201810767265566032837, 14.89955190132252276521065558170, 16.00428047524422166392652523124, 17.16385581487671333239807835693, 17.95109465790303748315278695912, 18.37352300430924408802329410860, 19.46426171165132255665891492843, 20.892751804824581803370960709950, 21.77952315635546099060122669415, 22.981660013580560497405517706877, 23.940902062269746320651493264216, 24.81386642045311394230013167408, 25.832263825179823757393678166884, 26.76234455440468268903738364286, 27.832280484973052113029356681475

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