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Properties

Label 1-41-41.25-r0-0-0
Degree $1$
Conductor $41$
Sign $0.991 + 0.129i$
Analytic cond. $0.190403$
Root an. cond. $0.190403$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.991 + 0.129i$
Analytic conductor: \(0.190403\)
Root analytic conductor: \(0.190403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ 0.991 + 0.129i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 - T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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