Properties

Degree 1
Conductor 199
Sign $0.682 - 0.731i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.991 + 0.126i)2-s + (−0.204 − 0.978i)3-s + (0.967 + 0.251i)4-s + (0.981 − 0.189i)5-s + (−0.0792 − 0.996i)6-s + (0.527 − 0.849i)7-s + (0.928 + 0.371i)8-s + (−0.916 + 0.400i)9-s + (0.997 − 0.0634i)10-s + (−0.959 − 0.281i)11-s + (0.0475 − 0.998i)12-s + (−0.605 + 0.795i)13-s + (0.630 − 0.776i)14-s + (−0.386 − 0.922i)15-s + (0.873 + 0.486i)16-s + (0.235 + 0.971i)17-s + ⋯
L(s,χ)  = 1  + (0.991 + 0.126i)2-s + (−0.204 − 0.978i)3-s + (0.967 + 0.251i)4-s + (0.981 − 0.189i)5-s + (−0.0792 − 0.996i)6-s + (0.527 − 0.849i)7-s + (0.928 + 0.371i)8-s + (−0.916 + 0.400i)9-s + (0.997 − 0.0634i)10-s + (−0.959 − 0.281i)11-s + (0.0475 − 0.998i)12-s + (−0.605 + 0.795i)13-s + (0.630 − 0.776i)14-s + (−0.386 − 0.922i)15-s + (0.873 + 0.486i)16-s + (0.235 + 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.682 - 0.731i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.682 - 0.731i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(199\)
\( \varepsilon \)  =  $0.682 - 0.731i$
motivic weight  =  \(0\)
character  :  $\chi_{199} (7, \cdot )$
Sato-Tate  :  $\mu(99)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 199,\ (0:\ ),\ 0.682 - 0.731i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.039807353 - 0.8868571668i$
$L(\frac12,\chi)$  $\approx$  $2.039807353 - 0.8868571668i$
$L(\chi,1)$  $\approx$  1.837982927 - 0.4995345596i
$L(1,\chi)$  $\approx$  1.837982927 - 0.4995345596i

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

−27.15735961738637393990272107796, −25.71560557298476196650460336994, −25.295955016305898372072713256300, −24.110720280601680125393140328958, −22.96125440401444420223627188899, −22.157138786648128104338853228582, −21.36893753989238872858034675349, −20.93338065569074939602280295200, −19.856418118410582514045048121765, −18.25071462983854044475364740266, −17.35247799571511828058594492565, −16.10303491087636772155737878031, −15.2069531609372792426961280659, −14.58586203653897895675856525972, −13.46007833412432305741466480584, −12.358009736207697401006636056001, −11.270774099446406016857537233884, −10.32239262984994791740548968704, −9.49903674068050353110330031491, −7.89401718988345614784168679548, −6.23119395783239934134981276350, −5.31666040327670954588498670132, −4.77212114930835113816138574531, −3.018741270012498302097045130126, −2.22528332269729869119086323575, 1.6127819628481458116308988224, 2.48149875239201959063710123709, 4.28114756611811162609557725819, 5.47214504323006157834184781238, 6.33704108437982012181282659496, 7.38008677303209962627263531717, 8.38158729874322022720552917365, 10.346352517810289168563094401578, 11.12336152754679230360885007543, 12.5572232283831549405520533356, 13.0233491713507005889761985261, 14.139305793467107678296204107926, 14.538828505583943368304469442775, 16.44475863819931097415845571317, 17.01992372218168430913781362109, 18.01583783177227936366098930180, 19.262173192590171804130242261506, 20.35601078895717269037852459330, 21.23485130548832837434193275308, 22.0573797483532831665168272514, 23.34714125332836781086394287134, 23.93621290651644985282421988953, 24.52586365582622866835581166286, 25.71308629654831591476537195368, 26.26878343349773329319813082292

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