Properties

Label 1-861-861.26-r1-0-0
Degree $1$
Conductor $861$
Sign $-0.971 + 0.238i$
Analytic cond. $92.5273$
Root an. cond. $92.5273$
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  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.994 + 0.104i)5-s + (−0.587 − 0.809i)8-s + (0.104 + 0.994i)10-s + (0.777 + 0.629i)11-s + (−0.453 − 0.891i)13-s + (0.669 − 0.743i)16-s + (−0.629 + 0.777i)17-s + (0.998 − 0.0523i)19-s + (−0.951 + 0.309i)20-s + (−0.453 + 0.891i)22-s + (−0.978 + 0.207i)23-s + (0.978 + 0.207i)25-s + (0.777 − 0.629i)26-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.994 + 0.104i)5-s + (−0.587 − 0.809i)8-s + (0.104 + 0.994i)10-s + (0.777 + 0.629i)11-s + (−0.453 − 0.891i)13-s + (0.669 − 0.743i)16-s + (−0.629 + 0.777i)17-s + (0.998 − 0.0523i)19-s + (−0.951 + 0.309i)20-s + (−0.453 + 0.891i)22-s + (−0.978 + 0.207i)23-s + (0.978 + 0.207i)25-s + (0.777 − 0.629i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.971 + 0.238i$
Analytic conductor: \(92.5273\)
Root analytic conductor: \(92.5273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 861,\ (1:\ ),\ -0.971 + 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2402504059 + 1.983239482i\)
\(L(\frac12)\) \(\approx\) \(0.2402504059 + 1.983239482i\)
\(L(1)\) \(\approx\) \(0.9687441024 + 0.7484347618i\)
\(L(1)\) \(\approx\) \(0.9687441024 + 0.7484347618i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.207 + 0.978i)T \)
5 \( 1 + (0.994 + 0.104i)T \)
11 \( 1 + (0.777 + 0.629i)T \)
13 \( 1 + (-0.453 - 0.891i)T \)
17 \( 1 + (-0.629 + 0.777i)T \)
19 \( 1 + (0.998 - 0.0523i)T \)
23 \( 1 + (-0.978 + 0.207i)T \)
29 \( 1 + (-0.987 + 0.156i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (0.838 + 0.544i)T \)
53 \( 1 + (-0.358 + 0.933i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.743 - 0.669i)T \)
67 \( 1 + (-0.933 - 0.358i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (0.0523 + 0.998i)T \)
97 \( 1 + (0.156 + 0.987i)T \)
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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

−21.50333792058093309467914825640, −20.76730135787071918379822429111, −19.99820204752299408315564487577, −19.24775279607964409552328359294, −18.32465440336402544387907584317, −17.76681287511713019592922439926, −16.85534830893800289645649107350, −16.02226368286928797757203810525, −14.565170546213450033840632880467, −14.02170143319745886980369912751, −13.537719849411726054747751423517, −12.454244558847023030775029214365, −11.73038446411653916684534269281, −10.95966358447139155218891926423, −9.96807409053981232205215107241, −9.24810471900876648882446331597, −8.78640633377064536592277373984, −7.25779012639723559361677768454, −6.15521509774798140970288907075, −5.3684758354388598592150871119, −4.421961297221828890077358512078, −3.42871278496386214363399606666, −2.32483186464691324776928157971, −1.5954896941571207043380032638, −0.41814765887809423453842402547, 1.164991677478349384578328179201, 2.44337435404854421473221636691, 3.68410631483293984019850690447, 4.65760231297660252932991766787, 5.66032212624575903488846883558, 6.20134110710375760725034766853, 7.21082162407438863437381118542, 7.95682726239010294263826611855, 9.161132825624276977105378670792, 9.63237284057850455061828021930, 10.552104175938757094362501444739, 11.924547431835485628836811693811, 12.7904994707352317060285213660, 13.478425528963613557993220558666, 14.2919079825865613192470265503, 14.97119369515358020421545590384, 15.69909088752470773949494718742, 16.83163898856496195788630807062, 17.36558385934340718859841020044, 17.92682278602995697863204488699, 18.74773935444506168532322670723, 19.96743213889221921801768942888, 20.67842905222808777537308036255, 21.95433502791511040369469321974, 22.17956314505252871120170391551

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