Properties

Degree $1$
Conductor $41$
Sign $0.991 - 0.129i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.809 + 0.587i)2-s − 3-s + (0.309 − 0.951i)4-s + (0.309 − 0.951i)5-s + (0.809 − 0.587i)6-s + (0.809 + 0.587i)7-s + (0.309 + 0.951i)8-s + 9-s + (0.309 + 0.951i)10-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)12-s + (0.809 − 0.587i)13-s − 14-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s,χ)  = 1  + (−0.809 + 0.587i)2-s − 3-s + (0.309 − 0.951i)4-s + (0.309 − 0.951i)5-s + (0.809 − 0.587i)6-s + (0.809 + 0.587i)7-s + (0.309 + 0.951i)8-s + 9-s + (0.309 + 0.951i)10-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)12-s + (0.809 − 0.587i)13-s − 14-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.991 - 0.129i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.991 - 0.129i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.991 - 0.129i$
Motivic weight: \(0\)
Character: $\chi_{41} (23, \cdot )$
Sato-Tate group: $\mu(10)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ 0.991 - 0.129i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.4873026749 - 0.03180323833i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.4873026749 - 0.03180323833i\)
\(L(\chi,1)\) \(\approx\) \(0.6114142044 + 0.01802249831i\)
\(L(1,\chi)\) \(\approx\) \(0.6114142044 + 0.01802249831i\)

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

−34.984254501804759587244567466409, −33.878197967781501676504288355507, −33.26918332820004703324056153375, −30.54246186127622718942952519722, −30.34890840544363748262373747328, −28.84373435213133995707631974566, −28.06889118214774505624792018605, −26.76632971469721988951922657744, −25.924023000009041982545646542827, −24.09851272117791975416883027727, −22.71116841331631779656826820935, −21.6162055757777312016604630552, −20.463700556156078473536953133442, −18.6993663167597326261276499363, −17.86986842840790771437802086752, −17.03141633997946244458859945163, −15.41809952229151813721660707541, −13.43495750858691407117555808344, −11.743833100515244573579243350604, −10.838393584040977891899984077177, −9.882080507589840565664365691085, −7.72331106798809226444182261884, −6.48594628866280505451603870731, −4.20563744319095472711933944425, −1.8360423828602710030063335269, 1.2867169775662419443004048703, 5.14454965864648021450586251474, 5.88981912823203962496981399464, 7.86651828064114693888874468392, 9.14436362234755681557024294112, 10.73110263523793203090085806061, 11.926004150168426676833888109468, 13.75413651416318800886651063257, 15.74271485854419867585598058357, 16.42044971765890086302721451705, 17.81347385252626295624997174794, 18.400326898442292499099275746959, 20.32572507880032566053539694079, 21.577211717757769605965832757152, 23.31431043403887928114793294971, 24.35236596859780341104600588701, 25.06257814859407242488634311536, 26.94757236094667513782322571590, 27.83895616686427845907724435300, 28.65823068505518347498396598374, 29.68277572481427833183376778321, 31.73876573392811167742403042537, 33.01828560938268431668244405330, 33.86609009308677022928535841149, 34.97003963865920722928513052337

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