Properties

Degree 1
Conductor 103
Sign $-0.947 + 0.319i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.982 + 0.183i)2-s + (−0.0922 − 0.995i)3-s + (0.932 − 0.361i)4-s + (0.602 + 0.798i)5-s + (0.273 + 0.961i)6-s + (−0.850 − 0.526i)7-s + (−0.850 + 0.526i)8-s + (−0.982 + 0.183i)9-s + (−0.739 − 0.673i)10-s + (0.982 − 0.183i)11-s + (−0.445 − 0.895i)12-s + (−0.850 − 0.526i)13-s + (0.932 + 0.361i)14-s + (0.739 − 0.673i)15-s + (0.739 − 0.673i)16-s + (−0.273 + 0.961i)17-s + ⋯
L(s,χ)  = 1  + (−0.982 + 0.183i)2-s + (−0.0922 − 0.995i)3-s + (0.932 − 0.361i)4-s + (0.602 + 0.798i)5-s + (0.273 + 0.961i)6-s + (−0.850 − 0.526i)7-s + (−0.850 + 0.526i)8-s + (−0.982 + 0.183i)9-s + (−0.739 − 0.673i)10-s + (0.982 − 0.183i)11-s + (−0.445 − 0.895i)12-s + (−0.850 − 0.526i)13-s + (0.932 + 0.361i)14-s + (0.739 − 0.673i)15-s + (0.739 − 0.673i)16-s + (−0.273 + 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.947 + 0.319i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.947 + 0.319i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(103\)
\( \varepsilon \)  =  $-0.947 + 0.319i$
motivic weight  =  \(0\)
character  :  $\chi_{103} (27, \cdot )$
Sato-Tate  :  $\mu(34)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 103,\ (1:\ ),\ -0.947 + 0.319i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.01783961245 - 0.1085791553i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.01783961245 - 0.1085791553i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5144025965 - 0.1146421639i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5144025965 - 0.1146421639i\)

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

−29.40852531638170070471178292641, −29.03272325486951216071308614432, −27.8882395044627494836384830711, −27.31169933502466970046220626624, −26.020062639016866019202271090, −25.30837628057776928758548474352, −24.33563389786112378822822690975, −22.35394229119225603552917737774, −21.723222583901961112391371391802, −20.479403688881910846893000625686, −19.87052010821713235279480595026, −18.50600871395143020818751965552, −17.10745215297782189670042186689, −16.57823729569423453526943410180, −15.65189602932557534957872447961, −14.26909553617596820855426721892, −12.42581926923324327778794228037, −11.601045135001387242777535879235, −9.83989806035153209892984269412, −9.56528623907787925293517483573, −8.55659897159850080475529980071, −6.667566557814188375711334975744, −5.39488754972682742076652258312, −3.67282535301774645533903898760, −2.0346881694698581852731368682, 0.06169825398935666173849898083, 1.74913736732033091216752771334, 3.06986652767887674701106982658, 5.988069279924308246771961183408, 6.694342199955469772397421791936, 7.622916913176868577538871706210, 9.14074364816620749085903351672, 10.248880085445063755110300213250, 11.35395054225252688463785087422, 12.6965521915913139503202112422, 13.99090442366061007288265135097, 15.05587807376766002447649647451, 16.7295902742104566721550323385, 17.44861600447158582208615722199, 18.31393189591131353132272850122, 19.63843092607289631275812395959, 19.73924320755251474116857068574, 21.81424776428107809471539605755, 22.843231019737476934820971250127, 24.15614366384306361914148149476, 25.00886341086808166890932557177, 25.95977522610591158440813084144, 26.60766965975673257646516628061, 28.10701933790723492220050995817, 29.07574803850608718234089201278

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