Properties

Degree $1$
Conductor $101$
Sign $0.985 + 0.170i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.876 + 0.481i)2-s + (0.992 − 0.125i)3-s + (0.535 − 0.844i)4-s + (0.876 + 0.481i)5-s + (−0.809 + 0.587i)6-s + (0.187 − 0.982i)7-s + (−0.0627 + 0.998i)8-s + (0.968 − 0.248i)9-s − 10-s + (−0.968 + 0.248i)11-s + (0.425 − 0.904i)12-s + (−0.187 − 0.982i)13-s + (0.309 + 0.951i)14-s + (0.929 + 0.368i)15-s + (−0.425 − 0.904i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s,χ)  = 1  + (−0.876 + 0.481i)2-s + (0.992 − 0.125i)3-s + (0.535 − 0.844i)4-s + (0.876 + 0.481i)5-s + (−0.809 + 0.587i)6-s + (0.187 − 0.982i)7-s + (−0.0627 + 0.998i)8-s + (0.968 − 0.248i)9-s − 10-s + (−0.968 + 0.248i)11-s + (0.425 − 0.904i)12-s + (−0.187 − 0.982i)13-s + (0.309 + 0.951i)14-s + (0.929 + 0.368i)15-s + (−0.425 − 0.904i)16-s + (−0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.985 + 0.170i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.985 + 0.170i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.985 + 0.170i$
Motivic weight: \(0\)
Character: $\chi_{101} (43, \cdot )$
Sato-Tate group: $\mu(50)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.985 + 0.170i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.036170861 + 0.08912721614i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.036170861 + 0.08912721614i\)
\(L(\chi,1)\) \(\approx\) \(1.039136687 + 0.1002765246i\)
\(L(1,\chi)\) \(\approx\) \(1.039136687 + 0.1002765246i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.70992808462503245421640162088, −28.64824366844807251741877257800, −28.027497761808979891662752390674, −26.44714715149205564145927884240, −26.081613285612851001020274978035, −24.863594528431841422221852571005, −24.29520815942851807608926270357, −21.76692348931841629733462163510, −21.364452989567056323458283644578, −20.47894509260579252478442801179, −19.23128407783151928179409035252, −18.480851289959087489960057436097, −17.35073262379491553290346444870, −16.06713699149771563451229744550, −15.03765326672115705727269818923, −13.43606911566493642433952336245, −12.66470354364917880296035918427, −11.08186724138045390254850907989, −9.75596024520057926150110477613, −8.935104078737103277855569748407, −8.1993369770296009419532719439, −6.53688089985247246787870340358, −4.620777412372634187980116010017, −2.65799355751165519662998722068, −1.95272867382749856251534942395, 1.6420593964706382511743934773, 2.99102510638002305528981887147, 5.18103551623355675409022429696, 6.87134288673926634587072616973, 7.631136877112663658714059111477, 8.876691302067114625225399325915, 10.14515943250716218450500640376, 10.650834646149841242095881296970, 13.01069336128386061293640310901, 14.02372288487951248650602057034, 14.91781245726843880810550215830, 16.03305465666497435642287827835, 17.53140873854603361900754508739, 18.122617015338931044378766695436, 19.317009264774369876017386304858, 20.38082352739017891622434861423, 21.074385336966127408409310456563, 22.88966929162386177310124817677, 24.0657016341470930961331346283, 25.1772980263647079508414897825, 25.75594096379849743331670154157, 26.71720651074206447674718824708, 27.37893070000854515444800601349, 29.18346467327602902555649619943, 29.60320335306546269157295868169

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