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Properties

Label 1-91-91.12-r1-0-0
Degree $1$
Conductor $91$
Sign $0.386 - 0.922i$
Analytic cond. $9.77930$
Root an. cond. $9.77930$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + 20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 6-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + 20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504706088 - 1.000903239i\)
\(L(\frac12)\) \(\approx\) \(1.504706088 - 1.000903239i\)
\(L(1)\) \(\approx\) \(1.306745977 - 0.1665233039i\)
\(L(1)\) \(\approx\) \(1.306745977 - 0.1665233039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−30.5690110672141188440946001971, −29.536826785317533869711824050461, −27.974243357232448226811620433973, −27.51935959451513790935647472949, −26.361742033861576574544071491763, −25.29235627332446817275733696687, −23.55277520209699859347871192793, −22.67500572061811618724928277744, −21.82086823465023602450306573730, −20.84733527722235905177800416574, −19.71972897390923796018419442845, −19.08580329650619040334706125765, −17.602528284687001081456508207374, −15.83691560578737825456209740063, −14.73429735760141850829193096301, −14.28238800842223436674824491468, −12.62461363938164221858114845924, −11.37035677219510489379470205458, −10.36900169137100635837378535378, −9.49872928602734950081577663438, −7.903412988647282315123237368973, −6.00467768010082878453741185086, −4.31762644950579175302745385805, −3.538639893272889006438356656219, −2.123672830375553716638849142377, 0.67381266013933904378462830049, 3.02493877661752143037337599248, 4.4828371817773077550661631318, 5.99432063024084267215745204359, 7.21917470510807733702447984152, 8.35816835836200369408634060784, 9.06593527613713344161429734749, 11.68298902263089414125491042482, 12.56657795587557223759438541188, 13.59942260220739947763125399939, 14.50428177625452688718919499606, 15.852181628259807512328984255227, 16.803229500088949914083945692626, 17.98515431639761199473215071212, 19.224149684770326792648040992042, 20.33699795265834074117345840631, 21.52463049554400950557045997504, 22.98551583422741841488553753855, 23.89370516666756552002359130641, 24.58103290893348022392675285393, 25.38318299131327214298723028351, 26.59233577611392797855597073238, 27.62231099111695430996623597136, 29.16288562876502989062581471393, 30.28151969593039861798781379048

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