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Properties

Label 1-133-133.86-r1-0-0
Degree $1$
Conductor $133$
Sign $-0.258 + 0.966i$
Analytic cond. $14.2928$
Root an. cond. $14.2928$
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  + (−0.766 − 0.642i)2-s + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (0.173 − 0.984i)5-s + (0.173 + 0.984i)6-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.766 + 0.642i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.173 − 0.984i)13-s + (−0.766 + 0.642i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + (0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (0.173 − 0.984i)5-s + (0.173 + 0.984i)6-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.766 + 0.642i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.173 − 0.984i)13-s + (−0.766 + 0.642i)15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.258 + 0.966i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ -0.258 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1986578534 - 0.2587020270i\)
\(L(\frac12)\) \(\approx\) \(-0.1986578534 - 0.2587020270i\)
\(L(1)\) \(\approx\) \(0.3547882779 - 0.3553022951i\)
\(L(1)\) \(\approx\) \(0.3547882779 - 0.3553022951i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−28.74504183514368514318565214647, −28.03378684954286657112962742845, −26.94106775152865568219735998656, −26.216997223340428140399877205196, −25.519125264474671351581595815131, −23.87715701386129625031846917084, −23.29133528982652101413914480248, −22.1903627238195044854740993253, −21.19500578773903892313458301011, −19.777500750158288881404235744878, −18.58183642382065312048742905781, −17.79366582249340502010297055520, −16.96352304239156553388086757166, −15.80955915126873575712101506958, −15.03080733166170500270304672019, −14.06982341380104597577863798878, −12.117453262412407599239044719582, −10.86198787217270702161828177287, −10.18717832928875051023445018532, −9.23296539276153449337494685824, −7.55457642526334992058606083868, −6.54986323620669505308992072907, −5.58280505070058914720979040360, −4.128220417018680593443079801449, −2.00346901352485149413154085320, 0.201987131893221278106500233852, 1.209021009728704343135912637979, 2.78812362470707168631759427869, 4.74862730674084547418092248278, 6.004026348950445649080495999258, 7.62267497398128350632701815399, 8.41057540578297726733309957174, 9.80980630483832482601132102398, 10.9119244672462201187726200439, 11.97589572145809585339903946158, 12.79994598560872433667108218538, 13.654461620761983083570861577347, 16.04019497024209063020770619751, 16.542168176866921846602528567045, 17.72640945167970335508993515627, 18.32788116181994504411561623932, 19.54563165901299756176749742418, 20.43064490070523201311944503946, 21.52583722863078291908501388336, 22.504247204596445576875268961850, 23.83519080765521179002726541675, 24.74989833860488912822839699263, 25.57628400403188800574387917932, 27.15798365972448041583659297729, 27.74812227966763575602320028688

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