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Properties

Label 1-91-91.68-r1-0-0
Degree $1$
Conductor $91$
Sign $0.372 + 0.927i$
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  + 2-s + (0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)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 + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)15-s + 16-s − 17-s + (−0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + ⋯
L(s)  = 1  + 2-s + (0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)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 + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)15-s + 16-s − 17-s + (−0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.372 + 0.927i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ 0.372 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.185169817 + 2.152677575i\)
\(L(\frac12)\) \(\approx\) \(3.185169817 + 2.152677575i\)
\(L(1)\) \(\approx\) \(2.201849028 + 0.8897386131i\)
\(L(1)\) \(\approx\) \(2.201849028 + 0.8897386131i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 - T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + T \)
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 \( 1 - T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 - T \)
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

−30.14323409487851962605475455765, −29.00836885419353172956241744854, −28.512329725298044463014254907321, −26.435751059244444170344574875881, −25.14348193379650195796396479640, −24.80107067538935635709829793928, −23.67071212076722136881394906350, −22.79389418205148003838985170998, −21.25131893580809716462670937615, −20.46660400366812620087533464274, −19.65083428305208075228529234089, −18.10359443200615760068803225816, −16.93286383364867861598435199314, −15.54507744765648408278361935759, −14.408110051760002344323523894415, −13.24098635216284888275285419524, −12.761119515408858404935414234993, −11.58554123658822047014260413099, −9.784429838982925856811192568208, −8.25471272790176430742713164309, −7.02488342849972649888469566433, −5.76150655215480074085753722798, −4.43968367265399756029011062582, −2.65973404623887319990438868594, −1.45192039226743102128059683980, 2.4517720946516190585517525120, 3.33940627049138566889065664762, 4.82289267221164018911124261548, 6.044245047082554872105438471456, 7.47131727576953797727931400384, 9.17796425479932213727581884040, 10.683656621625631311265756363557, 11.21024330242407472520948579449, 13.24596072516263036371477128258, 13.94594931374376557755212672838, 15.04980143752467422099538047690, 15.796881003737518329710841990615, 17.09901653871938785204110978337, 18.811102248824328253918665568039, 20.01433102596633927265335909602, 21.08862375806505150856754290570, 21.924189974655709448110963742645, 22.53444257613548027950542883059, 23.97185255193866820158581324015, 25.13816280095355668222057229669, 26.12519653283424559777193100263, 26.864447276891699556983964372963, 28.529415066925859400314164943802, 29.4729312121865532616338879916, 30.64398159855833291761241882882

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