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Properties

Label 2-1183-91.69-c0-0-4
Degree $2$
Conductor $1183$
Sign $0.982 - 0.188i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
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  + (1.07 + 0.623i)2-s + (0.277 + 0.480i)4-s + (0.866 − 0.5i)7-s − 0.554i·8-s + (−0.5 − 0.866i)9-s + (0.385 + 0.222i)11-s + 1.24·14-s + (0.623 − 1.07i)16-s − 1.24i·18-s + (0.277 + 0.480i)22-s + (−0.900 + 1.56i)23-s − 25-s + (0.480 + 0.277i)28-s + (−0.623 + 1.07i)29-s + (0.866 − 0.500i)32-s + ⋯
L(s)  = 1  + (1.07 + 0.623i)2-s + (0.277 + 0.480i)4-s + (0.866 − 0.5i)7-s − 0.554i·8-s + (−0.5 − 0.866i)9-s + (0.385 + 0.222i)11-s + 1.24·14-s + (0.623 − 1.07i)16-s − 1.24i·18-s + (0.277 + 0.480i)22-s + (−0.900 + 1.56i)23-s − 25-s + (0.480 + 0.277i)28-s + (−0.623 + 1.07i)29-s + (0.866 − 0.500i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.982 - 0.188i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ 0.982 - 0.188i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.855283928\)
\(L(\frac12)\) \(\approx\) \(1.855283928\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 \)
good2 \( 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1.56 - 0.900i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.80T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.56 + 0.900i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.56 + 0.900i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.775798865177692634249346235321, −9.320076033704814052375467321055, −7.996917772521239308750550439961, −7.43656034447779605522613982444, −6.38846904273901031860563634785, −5.81068269189107056406327273684, −4.85680940779112057054199183476, −4.04124681618615496508877426847, −3.27623261598262088157890472903, −1.46596527229273471752889255510, 2.01057104841784869057736722852, 2.60152763049168587404111067588, 3.99651480365822858977172744046, 4.58846499352521726903024710557, 5.59657994470432882962140859915, 6.09854552649976911640560776976, 7.72740003179484840527811201228, 8.199386164414392552465126615932, 9.056302490105213623858771084934, 10.23638503984602916216812181789

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