Properties

Degree 1
Conductor 103
Sign $0.155 + 0.987i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.696 + 0.717i)2-s + (−0.602 + 0.798i)3-s + (−0.0307 − 0.999i)4-s + (0.332 − 0.943i)5-s + (−0.153 − 0.988i)6-s + (0.213 + 0.976i)7-s + (0.739 + 0.673i)8-s + (−0.273 − 0.961i)9-s + (0.445 + 0.895i)10-s + (0.969 + 0.243i)11-s + (0.816 + 0.577i)12-s + (0.739 − 0.673i)13-s + (−0.850 − 0.526i)14-s + (0.552 + 0.833i)15-s + (−0.998 + 0.0615i)16-s + (−0.153 + 0.988i)17-s + ⋯
L(s,χ)  = 1  + (−0.696 + 0.717i)2-s + (−0.602 + 0.798i)3-s + (−0.0307 − 0.999i)4-s + (0.332 − 0.943i)5-s + (−0.153 − 0.988i)6-s + (0.213 + 0.976i)7-s + (0.739 + 0.673i)8-s + (−0.273 − 0.961i)9-s + (0.445 + 0.895i)10-s + (0.969 + 0.243i)11-s + (0.816 + 0.577i)12-s + (0.739 − 0.673i)13-s + (−0.850 − 0.526i)14-s + (0.552 + 0.833i)15-s + (−0.998 + 0.0615i)16-s + (−0.153 + 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.155 + 0.987i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.155 + 0.987i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(103\)
\( \varepsilon \)  =  $0.155 + 0.987i$
motivic weight  =  \(0\)
character  :  $\chi_{103} (63, \cdot )$
Sato-Tate  :  $\mu(51)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 103,\ (0:\ ),\ 0.155 + 0.987i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5072015110 + 0.4333904807i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5072015110 + 0.4333904807i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6311137131 + 0.3301776960i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6311137131 + 0.3301776960i\)

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

−29.58200158680727114078691803135, −28.78806570022772900258124570089, −27.58889013308488703993354191287, −26.6234288790596056564945689553, −25.67533758691681680624439337451, −24.50583792200904405857896605031, −23.08879822310167342508816585209, −22.34088408590319232095426027044, −21.19334466065449518485804690759, −19.81717844457458489825271907031, −18.98130706265351379427767379549, −18.00320995591027119019811185119, −17.28014657029970771517327420745, −16.261303271965227453573496563391, −14.06246193413050718352058105595, −13.4267537291101480571329253943, −11.80576602693029451981770447380, −11.133101164197886806794443537401, −10.17335792784497334549633772865, −8.6211843698317781117542629106, −7.124022928988896157885498957187, −6.55227881995540666946796758174, −4.25432643549321085250193546799, −2.580488003641404038378323277933, −1.11132400320711961466109563050, 1.46887222684878637771508002912, 4.21296674006912418363971648108, 5.59402069699954898549890622222, 6.14048754725439366281255749153, 8.27076874624456030114432637602, 9.088868368199085177250473360093, 10.05267675759020940309942485527, 11.38818304350074764212743439844, 12.640545643049927876439571117841, 14.44864872673061201516873716643, 15.44443259180231527833880973780, 16.29108317926612144365067104330, 17.30784724476381341858840158832, 17.95306650533432466392226185829, 19.48117284668329799253091762549, 20.661257153474976422268131482514, 21.712120889594854560176453702839, 22.91073479307735235863874104568, 24.00139832784753412699172786328, 25.06965690046935665439217640850, 25.76332459942718789685470175134, 27.31015131724595563463086356724, 27.91922126039197817671412505109, 28.42070422400346281892204652988, 29.60881183790652295449373163907

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