Properties

Label 2-1183-91.62-c0-0-4
Degree $2$
Conductor $1183$
Sign $-0.397 + 0.917i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.397 + 0.917i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.397 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08367349745\)
\(L(\frac12)\) \(\approx\) \(0.08367349745\)
\(L(1)\) 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

−9.711812020526006979000810327422, −8.765608831128056468011028369169, −8.040470758506595155701475052002, −7.46984382745816095153772812332, −6.55057356344604010028470829213, −5.84628755419726075911258865361, −4.54067656031988442557029525854, −3.48235677859109244194749211733, −2.12859959644883054663443318055, −0.099738193046379258498698453951, 1.68635329227123306944097875268, 2.91110653708313057372189274667, 3.70786929091803621800811469312, 5.43751721768950781468816140397, 5.92945203360945626997423483691, 7.12640575960103850587468187061, 8.058901751796472869728486958359, 8.875817890168211659299209883873, 9.477291430814399749751881996181, 9.940097183891417691129218610769

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