Properties

Degree 1
Conductor 199
Sign $0.374 - 0.927i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.605 − 0.795i)2-s + (0.950 − 0.312i)3-s + (−0.266 + 0.963i)4-s + (0.981 + 0.189i)5-s + (−0.823 − 0.567i)6-s + (0.472 − 0.881i)7-s + (0.928 − 0.371i)8-s + (0.805 − 0.592i)9-s + (−0.444 − 0.895i)10-s + (−0.959 + 0.281i)11-s + (0.0475 + 0.998i)12-s + (−0.386 + 0.922i)13-s + (−0.987 + 0.158i)14-s + (0.991 − 0.126i)15-s + (−0.857 − 0.513i)16-s + (0.235 − 0.971i)17-s + ⋯
L(s,χ)  = 1  + (−0.605 − 0.795i)2-s + (0.950 − 0.312i)3-s + (−0.266 + 0.963i)4-s + (0.981 + 0.189i)5-s + (−0.823 − 0.567i)6-s + (0.472 − 0.881i)7-s + (0.928 − 0.371i)8-s + (0.805 − 0.592i)9-s + (−0.444 − 0.895i)10-s + (−0.959 + 0.281i)11-s + (0.0475 + 0.998i)12-s + (−0.386 + 0.922i)13-s + (−0.987 + 0.158i)14-s + (0.991 − 0.126i)15-s + (−0.857 − 0.513i)16-s + (0.235 − 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.374 - 0.927i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.374 - 0.927i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(199\)
\( \varepsilon \)  =  $0.374 - 0.927i$
motivic weight  =  \(0\)
character  :  $\chi_{199} (70, \cdot )$
Sato-Tate  :  $\mu(99)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 199,\ (0:\ ),\ 0.374 - 0.927i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.129812546 - 0.7619392062i$
$L(\frac12,\chi)$  $\approx$  $1.129812546 - 0.7619392062i$
$L(\chi,1)$  $\approx$  1.094157519 - 0.5060927461i
$L(1,\chi)$  $\approx$  1.094157519 - 0.5060927461i

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

−26.79543752123387354403241852074, −26.08117610039468485716235741223, −25.202500048531807996359607259656, −24.708384092331354836453303837981, −23.78794425341448522195416988503, −22.178956005770476601639376177809, −21.34969198167326964194449351782, −20.41975150332165002080564486669, −19.27828976331344247894708086252, −18.35346053464989417486715604429, −17.59090262068419793606321032339, −16.44567837136304036366357045058, −15.226581560677851393989319740422, −14.89564771777284028412432661823, −13.62457849103589493106087025314, −12.86803623524651760918200192965, −10.74646429615032393541836114817, −9.992166954050652381988742212071, −8.86269504066364918614347821802, −8.35612040789090447677837282869, −7.12465366380328109119449897488, −5.55710534588359843052479331605, −4.992279116867438770437701555771, −2.837419808439254625077521221102, −1.70878090708323586927861324153, 1.45331369422162090211309478135, 2.3381286872163836831514664515, 3.53567389922116879520930295105, 4.92831810590061883240702828143, 7.04296865833950523526168124079, 7.67158711480318630883215546717, 8.990121169512878024848481051075, 9.81842806007798656117764056170, 10.61138499316471406980845007149, 11.98146602267889048631005860886, 13.2766279756002848075153634481, 13.72244312560244048481088289555, 14.78685178205880621932646111966, 16.47122574763579256592650979241, 17.37272090310831466294934394811, 18.45645651898912205819149928845, 18.891987593892179876247351771395, 20.39697899795228867897166262213, 20.70774914302781014376921370810, 21.50038183071544380356132684528, 22.81495110264985563309371844546, 24.11419437143210890016896783151, 25.19328867543058046968881949043, 26.01839884164575450914669186758, 26.62454911359342626300971219402

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