Properties

Label 1-231-231.215-r1-0-0
Degree $1$
Conductor $231$
Sign $-0.913 + 0.406i$
Analytic cond. $24.8243$
Root an. cond. $24.8243$
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  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.104 + 0.994i)5-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.809 + 0.587i)13-s + (−0.104 + 0.994i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (−0.809 + 0.587i)20-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (−0.978 + 0.207i)26-s + (0.309 − 0.951i)29-s + (0.104 + 0.994i)31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.104 + 0.994i)5-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.809 + 0.587i)13-s + (−0.104 + 0.994i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (−0.809 + 0.587i)20-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (−0.978 + 0.207i)26-s + (0.309 − 0.951i)29-s + (0.104 + 0.994i)31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.913 + 0.406i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ -0.913 + 0.406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5262138513 + 2.475367982i\)
\(L(\frac12)\) \(\approx\) \(0.5262138513 + 2.475367982i\)
\(L(1)\) \(\approx\) \(1.306722119 + 0.9761524251i\)
\(L(1)\) \(\approx\) \(1.306722119 + 0.9761524251i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−25.21535697575084012329701871777, −24.607314685256516469272094786040, −23.91016061230092799880193868826, −22.782137946787675584369042533954, −22.09751807126861772120276667888, −20.93377215381518374364664068066, −20.28041016389889782436988909408, −19.576016137468696165170148314291, −18.33163033188226722914258258530, −16.991069284376686482726756956557, −16.07500216144376951130101548069, −15.163864488491284720749979442543, −14.10758035964675971370476905945, −13.06531691947367720550651881930, −12.39026627414633648717648068044, −11.478648063116968380542439713380, −10.267214851053665206240768535541, −9.24377181259322075560669230886, −7.88830395031569432635521070796, −6.606922181426408599513411267168, −5.280339785603739870258491200321, −4.65171458607979231777598863290, −3.33034649075870518761525844957, −1.97866233891074124861060652388, −0.56557745107751642036379597510, 2.13908552001428663356448154901, 3.183839980051404248796007802090, 4.347251331882062937458517447280, 5.52877902282617219956772423076, 6.82530089524662811991232361908, 7.27041507215193749680779487783, 8.721105185196542723432734710591, 10.22028794901005322298343960403, 11.33944167378993587812644338702, 12.04349992463907621377312742152, 13.42356908313243486493977060068, 14.07583393322503405020984363550, 15.16725765307872284901681159427, 15.67606113186763873722596621397, 17.06333538218597644186949052459, 17.81333680027729187205877681980, 19.19785216833991372617173508951, 19.9916327326825677788112022537, 21.36804757159077676774645014479, 21.9615314275207007077446478743, 22.75956911935067924667311190887, 23.68369369955921099135092025391, 24.50906124173315967258416584117, 25.511917330348223863654003986289, 26.44802151899878278395608210332

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