↑

Properties

Label 1-167-167.18-r0-0-0
Degree $1$
Conductor $167$
Sign $0.950 + 0.311i$
Analytic cond. $0.775544$
Root an. cond. $0.775544$
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  + (0.929 − 0.369i)2-s + (0.776 + 0.629i)3-s + (0.726 − 0.686i)4-s + (−0.700 + 0.713i)5-s + (0.954 + 0.298i)6-s + (0.898 + 0.438i)7-s + (0.421 − 0.906i)8-s + (0.206 + 0.978i)9-s + (−0.387 + 0.922i)10-s + (−0.965 + 0.261i)11-s + (0.997 − 0.0756i)12-s + (−0.982 + 0.188i)13-s + (0.997 + 0.0756i)14-s + (−0.993 + 0.113i)15-s + (0.0567 − 0.998i)16-s + (0.280 − 0.959i)17-s + ⋯
L(s)  = 1  + (0.929 − 0.369i)2-s + (0.776 + 0.629i)3-s + (0.726 − 0.686i)4-s + (−0.700 + 0.713i)5-s + (0.954 + 0.298i)6-s + (0.898 + 0.438i)7-s + (0.421 − 0.906i)8-s + (0.206 + 0.978i)9-s + (−0.387 + 0.922i)10-s + (−0.965 + 0.261i)11-s + (0.997 − 0.0756i)12-s + (−0.982 + 0.188i)13-s + (0.997 + 0.0756i)14-s + (−0.993 + 0.113i)15-s + (0.0567 − 0.998i)16-s + (0.280 − 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.950 + 0.311i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.950 + 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.193809403 + 0.3498121089i\)
\(L(\frac12)\) \(\approx\) \(2.193809403 + 0.3498121089i\)
\(L(1)\) \(\approx\) \(1.953524841 + 0.1503256558i\)
\(L(1)\) \(\approx\) \(1.953524841 + 0.1503256558i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (0.929 - 0.369i)T \)
3 \( 1 + (0.776 + 0.629i)T \)
5 \( 1 + (-0.700 + 0.713i)T \)
7 \( 1 + (0.898 + 0.438i)T \)
11 \( 1 + (-0.965 + 0.261i)T \)
13 \( 1 + (-0.982 + 0.188i)T \)
17 \( 1 + (0.280 - 0.959i)T \)
19 \( 1 + (-0.521 - 0.853i)T \)
23 \( 1 + (0.988 - 0.150i)T \)
29 \( 1 + (0.988 + 0.150i)T \)
31 \( 1 + (-0.752 - 0.658i)T \)
37 \( 1 + (0.206 - 0.978i)T \)
41 \( 1 + (0.132 + 0.991i)T \)
43 \( 1 + (-0.999 + 0.0378i)T \)
47 \( 1 + (-0.316 + 0.948i)T \)
53 \( 1 + (-0.243 - 0.969i)T \)
59 \( 1 + (0.280 + 0.959i)T \)
61 \( 1 + (-0.584 - 0.811i)T \)
67 \( 1 + (-0.700 - 0.713i)T \)
71 \( 1 + (0.351 - 0.936i)T \)
73 \( 1 + (0.0567 + 0.998i)T \)
79 \( 1 + (-0.169 + 0.985i)T \)
83 \( 1 + (0.929 + 0.369i)T \)
89 \( 1 + (-0.993 - 0.113i)T \)
97 \( 1 + (-0.752 + 0.658i)T \)
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

−27.264912192101563942665256154255, −26.525334069027355439072942319083, −25.27402588197258237339962410651, −24.55870100560055398985205110423, −23.58674900024305925318231666656, −23.43290777614629818146113496852, −21.530680585322089319765567108486, −20.80176493024096070341793288974, −20.022818983377640346127125053038, −19.035437145278011360954418219756, −17.53109734732081575510572766144, −16.61451213375647634369902243471, −15.25737353456626316926024509136, −14.699502460755610478484166736956, −13.5522741726757975672002546304, −12.67142600712457716944183249314, −11.94220624379364730125079613263, −10.5417364318480411461591417131, −8.47979363392121242357999191113, −7.953927918955581132518415961675, −7.055179378604085933079750104032, −5.392447883553606661581054329617, −4.33680218994191790562176807049, −3.14371331349879424910745650023, −1.66629451810705721470293693961, 2.34417861806818521060992481509, 2.97460882674934230709549791663, 4.50396396404550072413520015707, 5.10118552236474304491420122463, 7.05934532825637060080642910089, 7.95646890041990622710366059565, 9.52223140835010913374590461243, 10.71422788579733556461905773504, 11.41323357776419513521865947761, 12.66952133494276370643521086051, 13.94523259873556533140083704486, 14.85835934736343923314102978085, 15.23946056662337358564241992475, 16.31207302555611065936849681783, 18.19876200307534099129695276440, 19.21915002597340499822179966230, 20.04095526104719512511218942278, 21.07528650688766545716430807752, 21.68465495265400666667340583514, 22.70529509153329133790546134901, 23.71135167698918403971647663533, 24.68280868969055065738905416135, 25.6786473379517947830109407985, 26.86757749818302199241011387932, 27.55661942539607292822374731163

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