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Properties

Label 1-199-199.70-r0-0-0
Degree $1$
Conductor $199$
Sign $0.374 - 0.927i$
Analytic cond. $0.924152$
Root an. cond. $0.924152$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.374 - 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129812546 - 0.7619392062i\)
\(L(\frac12)\) \(\approx\) \(1.129812546 - 0.7619392062i\)
\(L(1)\) \(\approx\) \(1.094157519 - 0.5060927461i\)
\(L(1)\) \(\approx\) \(1.094157519 - 0.5060927461i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.605 - 0.795i)T \)
3 \( 1 + (0.950 - 0.312i)T \)
5 \( 1 + (0.981 + 0.189i)T \)
7 \( 1 + (0.472 - 0.881i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (-0.386 + 0.922i)T \)
17 \( 1 + (0.235 - 0.971i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.967 + 0.251i)T \)
29 \( 1 + (-0.975 + 0.220i)T \)
31 \( 1 + (-0.204 - 0.978i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (-0.916 - 0.400i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.527 + 0.849i)T \)
53 \( 1 + (0.991 + 0.126i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (-0.987 - 0.158i)T \)
73 \( 1 + (-0.701 + 0.712i)T \)
79 \( 1 + (0.873 - 0.486i)T \)
83 \( 1 + (0.0475 - 0.998i)T \)
89 \( 1 + (0.678 + 0.734i)T \)
97 \( 1 + (-0.745 + 0.666i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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