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Properties

Label 2-1166-53.52-c1-0-25
Degree $2$
Conductor $1166$
Sign $0.299 + 0.954i$
Analytic cond. $9.31055$
Root an. cond. $3.05132$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + 3.24i·3-s − 4-s − 1.22i·5-s + 3.24·6-s − 3.83·7-s + i·8-s − 7.56·9-s − 1.22·10-s − 11-s − 3.24i·12-s − 1.39·13-s + 3.83i·14-s + 3.96·15-s + 16-s + 7.82·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.87i·3-s − 0.5·4-s − 0.546i·5-s + 1.32·6-s − 1.44·7-s + 0.353i·8-s − 2.52·9-s − 0.386·10-s − 0.301·11-s − 0.938i·12-s − 0.387·13-s + 1.02i·14-s + 1.02·15-s + 0.250·16-s + 1.89·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1166\)    =    \(2 \cdot 11 \cdot 53\)
Sign: $0.299 + 0.954i$
Analytic conductor: \(9.31055\)
Root analytic conductor: \(3.05132\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1166} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1166,\ (\ :1/2),\ 0.299 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7068657154\)
\(L(\frac12)\) \(\approx\) \(0.7068657154\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
11 \( 1 + T \)
53 \( 1 + (-6.94 + 2.17i)T \)
good3 \( 1 - 3.24iT - 3T^{2} \)
5 \( 1 + 1.22iT - 5T^{2} \)
7 \( 1 + 3.83T + 7T^{2} \)
13 \( 1 + 1.39T + 13T^{2} \)
17 \( 1 - 7.82T + 17T^{2} \)
19 \( 1 + 6.90iT - 19T^{2} \)
23 \( 1 - 5.07iT - 23T^{2} \)
29 \( 1 - 0.736T + 29T^{2} \)
31 \( 1 + 5.86iT - 31T^{2} \)
37 \( 1 + 7.76T + 37T^{2} \)
41 \( 1 + 11.4iT - 41T^{2} \)
43 \( 1 + 2.75T + 43T^{2} \)
47 \( 1 - 9.74T + 47T^{2} \)
59 \( 1 + 8.71T + 59T^{2} \)
61 \( 1 - 1.83iT - 61T^{2} \)
67 \( 1 + 6.72iT - 67T^{2} \)
71 \( 1 - 7.31iT - 71T^{2} \)
73 \( 1 + 8.08iT - 73T^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 + 4.35iT - 83T^{2} \)
89 \( 1 + 18.7T + 89T^{2} \)
97 \( 1 - 7.53T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.616790831926210558829082647880, −9.281457652682695218520470568880, −8.513072890113602112658243426148, −7.19180550394483191778624590758, −5.64305818319870796432158204922, −5.26998405161923890523420948722, −4.20214206937547529744687837898, −3.39247528565908932155712604310, −2.77901829492811735834555508756, −0.34463163216317546763480168101, 1.16961211942500447333347494717, 2.75728682372272547594239633683, 3.42081613035913207206519928226, 5.37279731980507275177392948181, 6.11821147475848170307763740229, 6.66950553533917378734837442909, 7.34091106124770681594459785402, 7.985966619775992914109760945608, 8.778036198503954696310757511195, 9.949559078620426086260649208652

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