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Properties

Label 1-91-91.81-r0-0-0
Degree $1$
Conductor $91$
Sign $0.949 - 0.313i$
Analytic cond. $0.422602$
Root an. cond. $0.422602$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.949 - 0.313i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ 0.949 - 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9767332815 - 0.1570047847i\)
\(L(\frac12)\) \(\approx\) \(0.9767332815 - 0.1570047847i\)
\(L(1)\) \(\approx\) \(1.007004944 - 0.1850325297i\)
\(L(1)\) \(\approx\) \(1.007004944 - 0.1850325297i\)

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

−30.85631996721330325406398610194, −29.28550263641960015192559110287, −27.91642929182493223779743583195, −27.19696006923612081757532201971, −26.28566806785960122381797701832, −24.96022220044995825341700544501, −24.636314190821477060976063758042, −23.48988133027482021892352098121, −22.09512358632186745575945271824, −20.396725951734820359894869111, −19.78639284733268161014351845496, −18.738811134799662320325179915585, −17.42904048103422340368483565436, −16.17406257215225960603197924301, −15.45726837615807741872003360481, −14.21564087234651171507650128552, −13.320483719207626187642652399272, −11.70883935797959904241456037316, −9.7240567297117731430235552385, −9.00484139671878300585789558416, −7.966002729916630832123393169242, −6.92289323215062476602763491594, −5.093098890930989372877598680348, −3.79956034922977328568847244800, −1.446201272130336290818164343048, 1.86072577829427776750944084824, 3.2570357091673899810630088535, 4.128602173910420490650296022107, 6.8759758624459791421223462569, 8.00638098561502354126553336588, 9.111651989947304782493824396990, 10.24320989333641488061747342089, 11.404991008550216376003018108532, 12.63255435249446604218454995955, 13.961384217883075236715592950352, 14.88357318229387502295620342180, 16.325864533339061761476664533811, 17.89849431826216230984717117171, 18.83292899597569550000111389582, 19.69661006944660717484403185547, 20.40525383209390842499222151287, 21.82685904757055225667125348364, 22.47874546847502890756775046861, 24.18006438071847568359474589319, 25.53211042315779596955106311271, 26.41263179424461789850551828357, 27.0787018841727903376592047879, 28.11898218664083886633387439860, 29.60890926828456603817894168988, 30.442679083258150633502426726675

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