Properties

Degree 1
Conductor 199
Sign $-0.588 + 0.808i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.472 − 0.881i)2-s + (−0.823 + 0.567i)3-s + (−0.553 − 0.832i)4-s + (0.0475 − 0.998i)5-s + (0.110 + 0.993i)6-s + (−0.701 + 0.712i)7-s + (−0.995 + 0.0950i)8-s + (0.356 − 0.934i)9-s + (−0.857 − 0.513i)10-s + (−0.654 + 0.755i)11-s + (0.928 + 0.371i)12-s + (−0.999 + 0.0317i)13-s + (0.296 + 0.954i)14-s + (0.527 + 0.849i)15-s + (−0.386 + 0.922i)16-s + (−0.327 − 0.945i)17-s + ⋯
L(s,χ)  = 1  + (0.472 − 0.881i)2-s + (−0.823 + 0.567i)3-s + (−0.553 − 0.832i)4-s + (0.0475 − 0.998i)5-s + (0.110 + 0.993i)6-s + (−0.701 + 0.712i)7-s + (−0.995 + 0.0950i)8-s + (0.356 − 0.934i)9-s + (−0.857 − 0.513i)10-s + (−0.654 + 0.755i)11-s + (0.928 + 0.371i)12-s + (−0.999 + 0.0317i)13-s + (0.296 + 0.954i)14-s + (0.527 + 0.849i)15-s + (−0.386 + 0.922i)16-s + (−0.327 − 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.588 + 0.808i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.588 + 0.808i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(199\)
\( \varepsilon \)  =  $-0.588 + 0.808i$
motivic weight  =  \(0\)
character  :  $\chi_{199} (26, \cdot )$
Sato-Tate  :  $\mu(99)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 199,\ (0:\ ),\ -0.588 + 0.808i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.03403130589 - 0.06681840322i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.03403130589 - 0.06681840322i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5449986802 - 0.2654772429i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5449986802 - 0.2654772429i\)

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

−27.188104039869079515019305076415, −26.37173470162585663891936720230, −25.73862678433147992383248312130, −24.350307571361432188319476746398, −23.803237000704087652281953325499, −22.88568225806304714887009980246, −22.16539055712697254018989661020, −21.51959682907865559916214885098, −19.5729549414837687057688093665, −18.7578920437680596781027173669, −17.62333558481034974146157315375, −17.06028127966772608519792259662, −15.95372688420515594138550141134, −15.05252907849202784974923903698, −13.624375428925367613027711670319, −13.29379372109384500251870008367, −11.968971049978095167371684893703, −10.87584992659499503622905801970, −9.78501943858934299915638125283, −7.932408472969759293015130118451, −7.15266443080799948708707149001, −6.34717761710599839981247560691, −5.42719540360956119922318221594, −3.968110900179859104521166876902, −2.63103079646095196374554260365, 0.05092049587488900405022201817, 1.976585042203733964849844079432, 3.49102444945472385669717447973, 5.01346967073774581766086187594, 5.15819969702832067782256854476, 6.66180626356991426407209609256, 8.72628171084719820799258105069, 9.7506256528126599062798090495, 10.29145609659914151928564517177, 11.96684101866968779980737279167, 12.22362315289366464824378255267, 13.148568194247380526856889954333, 14.66687960188438184393469910014, 15.71013937777053982951469665786, 16.52488742433460850568011358079, 17.810835076044201922653512587740, 18.63226370423707391274674700115, 20.02797245034997127902635872654, 20.64384795576798070275856325594, 21.64212689535742429402746412244, 22.39132245171117839151025478664, 23.17345065075412663410779834697, 24.15861549031963654913585051304, 25.1491690435380930540293982402, 26.728598432445759382264045544829

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