Properties

Degree 1
Conductor 167
Sign $0.950 + 0.311i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(167\)
\( \varepsilon \)  =  $0.950 + 0.311i$
motivic weight  =  \(0\)
character  :  $\chi_{167} (18, \cdot )$
Sato-Tate  :  $\mu(83)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 167,\ (0:\ ),\ 0.950 + 0.311i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.193809403 + 0.3498121089i$
$L(\frac12,\chi)$  $\approx$  $2.193809403 + 0.3498121089i$
$L(\chi,1)$  $\approx$  1.953524841 + 0.1503256558i
$L(1,\chi)$  $\approx$  1.953524841 + 0.1503256558i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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