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Properties

Label 1-91-91.62-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: $\chi_{91} (62, \cdot )$
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

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

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