Properties

Degree 1
Conductor 199
Sign $-0.922 - 0.386i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.950 − 0.312i)2-s + (−0.266 − 0.963i)3-s + (0.805 − 0.592i)4-s + (−0.888 − 0.458i)5-s + (−0.553 − 0.832i)6-s + (−0.823 + 0.567i)7-s + (0.580 − 0.814i)8-s + (−0.857 + 0.513i)9-s + (−0.987 − 0.158i)10-s + (−0.654 − 0.755i)11-s + (−0.786 − 0.618i)12-s + (−0.745 − 0.666i)13-s + (−0.605 + 0.795i)14-s + (−0.204 + 0.978i)15-s + (0.296 − 0.954i)16-s + (0.981 − 0.189i)17-s + ⋯
L(s,χ)  = 1  + (0.950 − 0.312i)2-s + (−0.266 − 0.963i)3-s + (0.805 − 0.592i)4-s + (−0.888 − 0.458i)5-s + (−0.553 − 0.832i)6-s + (−0.823 + 0.567i)7-s + (0.580 − 0.814i)8-s + (−0.857 + 0.513i)9-s + (−0.987 − 0.158i)10-s + (−0.654 − 0.755i)11-s + (−0.786 − 0.618i)12-s + (−0.745 − 0.666i)13-s + (−0.605 + 0.795i)14-s + (−0.204 + 0.978i)15-s + (0.296 − 0.954i)16-s + (0.981 − 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.922 - 0.386i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.922 - 0.386i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(199\)
\( \varepsilon \)  =  $-0.922 - 0.386i$
motivic weight  =  \(0\)
character  :  $\chi_{199} (45, \cdot )$
Sato-Tate  :  $\mu(99)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 199,\ (0:\ ),\ -0.922 - 0.386i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2365652155 - 1.177299545i$
$L(\frac12,\chi)$  $\approx$  $0.2365652155 - 1.177299545i$
$L(\chi,1)$  $\approx$  0.8945350934 - 0.8076845871i
$L(1,\chi)$  $\approx$  0.8945350934 - 0.8076845871i

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

−27.074840508715150335095095816, −26.232629186560133311192070935, −25.753339359877505752440960014441, −24.13649790824104950739317948617, −23.25107962960497419090575181450, −22.68042291620797092708432643922, −21.98953512933025107814546927919, −20.72857952688545706697433482682, −20.12547777094062627484739262025, −18.93848387489213352850910894237, −17.2607948464099595229963442749, −16.32225497272969387301675776547, −15.76477184860237760914533998340, −14.77863325258680841637557916474, −14.01900244710098127147625886804, −12.45033353596855881333475994647, −11.82885048715495829219550436206, −10.5726510488370380284864665915, −9.79205984892566323875535197590, −7.93034988168686374003697263530, −7.04721229226693846623368275178, −5.82433000522090933932555546791, −4.552306996885037116538577077896, −3.78211656940516847417246692477, −2.78287044929574143705457794042, 0.68171199597909241924454745870, 2.54053743982569113363037204695, 3.44648051576053261745759409198, 5.19112967885928430396867966213, 5.85240879006108513497026531583, 7.23663062561726668605557026995, 8.063372526515372251602227468523, 9.74959748701989153275711219384, 11.22472540142146040282923825303, 11.98841599815005198901539042116, 12.74413242279260062991630990252, 13.44302703720491715316875786245, 14.72394267203818069261589004554, 15.88497458813117111064610689726, 16.49528140220238820220031918198, 18.15037853815331494627850342381, 19.27876303408340170708663143846, 19.60473332546216071163428670681, 20.77231963588917486298470210618, 22.08934149221825770105566393199, 22.79479327784092047687832449814, 23.64829399823996977500061989240, 24.414552952982287059443042934390, 25.06780736343510497655044325694, 26.34041388772768174678127067066

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