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Properties

Label 1-91-91.82-r1-0-0
Degree $1$
Conductor $91$
Sign $0.113 + 0.993i$
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 − 3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s − 8-s + 9-s − 10-s − 11-s + (0.5 + 0.866i)12-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + 19-s + (−0.5 + 0.866i)20-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s − 8-s + 9-s − 10-s − 11-s + (0.5 + 0.866i)12-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + 19-s + (−0.5 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.113 + 0.993i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ 0.113 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03690587958 - 0.03292383618i\)
\(L(\frac12)\) \(\approx\) \(-0.03690587958 - 0.03292383618i\)
\(L(1)\) \(\approx\) \(0.5292775333 - 0.3920383368i\)
\(L(1)\) \(\approx\) \(0.5292775333 - 0.3920383368i\)

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 \( 1 - T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 - T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + (-0.5 + 0.866i)T \)
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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

−30.99792089199533857874101832462, −30.0765175024440604939062994102, −28.97497651474324900364520104584, −27.53220107303560781624533469841, −26.73644438010649365804329976385, −25.74191397266784433499816982990, −24.29240410734129635283490683938, −23.522697217022547648955030178631, −22.61957386670859592505612733227, −21.99354691991791607131419565351, −20.63360128713030902205644809947, −18.447054672104874240273537019265, −18.20776718624007391462040528876, −16.62090923024176382533142092586, −15.87676240738494914000785509526, −14.8389546340914285148958638364, −13.52099444324612597211000736665, −12.25646937156844535097945709716, −11.25491669000972543842404174808, −9.883822846704352633448833210665, −7.86026737991244930719531803925, −7.01744807601371376033453947665, −5.797091838674530570792988030, −4.66497683571892600161296838549, −3.11451161786997496841547581693, 0.02250401116457953775278419475, 1.50210023976149264941858508070, 3.680639668364564465362900767529, 4.97410524842875395822203753926, 5.787156274838776957966996972034, 7.76996382707392668740752133156, 9.498716921520984122933475000947, 10.64781797775942713110043727080, 11.74379720687314163472326933332, 12.56107895857648874710287768474, 13.48620059791671230853499119415, 15.28622437516863153614186852465, 16.24851741594278671203144427637, 17.619547169384583055662006636544, 18.69076682928060915672528934482, 19.86239273525843448580467292497, 20.993588563770209546182644473706, 21.80228868209287068530717676560, 23.134989484114658795527943198890, 23.659215136921814155252962823282, 24.604579912756975848775445726926, 26.639048429985379291754807526453, 27.775297805722295640459749074437, 28.42937220469573332660184061406, 29.16107286045576698404465053973

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