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Properties

Label 1-91-91.69-r1-0-0
Degree $1$
Conductor $91$
Sign $-0.872 + 0.488i$
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 + 5-s + (−0.5 + 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s − 12-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s − 18-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s − 12-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s − 18-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ -0.872 + 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6813508533 + 2.609652806i\)
\(L(\frac12)\) \(\approx\) \(0.6813508533 + 2.609652806i\)
\(L(1)\) \(\approx\) \(1.123840881 + 1.331834459i\)
\(L(1)\) \(\approx\) \(1.123840881 + 1.331834459i\)

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 \( 1 + T \)
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 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−29.994580129461080602783308011472, −29.01755162895180494066125252783, −28.0247305332071904230749751411, −26.493677101426294081394463970197, −25.3561444961999817071585296500, −24.29482907538213393389654220923, −23.47300795306202113393727389897, −21.97517973452235100286382775389, −21.286216603732683781593933709295, −20.0298733174233671670960426395, −19.17261307749634275518427305872, −18.1734430847660527848162712754, −17.116842498019459121984754810783, −14.98627206640684817040840943527, −13.961771547368313593246245254333, −13.30439322112523550249665488297, −12.24381701358986586137001854052, −10.9477788933524784324796306004, −9.553815683042859715089513427082, −8.53640684290918454328229588929, −6.55523371314234657369206556524, −5.566528366071156947613805044489, −3.59114960438512735565631043357, −2.28118723537591261063672022227, −1.08282524070479102740216535629, 2.49257831927039144995483219673, 4.08343010469098027018670366350, 5.201776811051564669472604835128, 6.46352852165912019975439563887, 8.01873158919181653531694813786, 9.28562901902858132663039594425, 10.13841531336163721827638831998, 12.09285570150393887674052258235, 13.51493937663361020833555914927, 14.356575387057077664942614668708, 15.19224106037564380509587116894, 16.502979443249322331760509600715, 17.22764931674643657867169395645, 18.55626922313008652444352661710, 20.44150388518792635812044658407, 21.12963386986081349176743498934, 22.24241625465255262254411844636, 22.93137527164515289627098378912, 24.68900877300146614809503373187, 25.31651536047684345917145978861, 26.12781301269379487243396521889, 27.17158868737343647816726700964, 28.26214029648913688555008551480, 29.8429095108616579861162364690, 30.83624510150130346112163970102

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