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Properties

Label 2-1183-7.4-c1-0-89
Degree $2$
Conductor $1183$
Sign $0.581 - 0.813i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
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  + (−0.680 − 1.17i)2-s + (0.656 − 1.13i)3-s + (0.0732 − 0.126i)4-s + (−1.52 − 2.64i)5-s − 1.78·6-s + (−2.33 + 1.24i)7-s − 2.92·8-s + (0.638 + 1.10i)9-s + (−2.08 + 3.60i)10-s + (0.775 − 1.34i)11-s + (−0.0961 − 0.166i)12-s + (3.05 + 1.90i)14-s − 4.01·15-s + (1.84 + 3.19i)16-s + (−2.91 + 5.04i)17-s + (0.868 − 1.50i)18-s + ⋯
L(s)  = 1  + (−0.481 − 0.833i)2-s + (0.379 − 0.656i)3-s + (0.0366 − 0.0634i)4-s + (−0.683 − 1.18i)5-s − 0.729·6-s + (−0.882 + 0.470i)7-s − 1.03·8-s + (0.212 + 0.368i)9-s + (−0.658 + 1.14i)10-s + (0.233 − 0.404i)11-s + (−0.0277 − 0.0480i)12-s + (0.816 + 0.509i)14-s − 1.03·15-s + (0.460 + 0.797i)16-s + (−0.706 + 1.22i)17-s + (0.204 − 0.354i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08054455061\)
\(L(\frac12)\) \(\approx\) \(0.08054455061\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.33 - 1.24i)T \)
13 \( 1 \)
good2 \( 1 + (0.680 + 1.17i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.656 + 1.13i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.52 + 2.64i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.775 + 1.34i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.91 - 5.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.722 - 1.25i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.13 + 5.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.88T + 29T^{2} \)
31 \( 1 + (-0.763 + 1.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.87 - 6.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.17T + 41T^{2} \)
43 \( 1 - 5.03T + 43T^{2} \)
47 \( 1 + (-2.60 - 4.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.88 - 6.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.39 + 2.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.87 + 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.341 - 0.591i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.582T + 71T^{2} \)
73 \( 1 + (-2.16 + 3.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.20 + 5.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + (1.03 + 1.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.78T + 97T^{2} \)
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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

−8.981511475613301109774270255561, −8.497260762470126243396360595396, −7.76344451183582387611975943421, −6.49375788341902048087175244759, −5.84993678107918252322533368856, −4.52870156862466247312669971534, −3.48890193961561711857835183650, −2.30846575197043885509358163262, −1.38345790844367407269494787044, −0.03858233658829970527323771731, 2.69213675102637470759761565364, 3.50928317870834399515738634616, 4.09332360555229168056808272508, 5.68581176325863275354410917343, 6.80326763996149224781208000998, 7.10544165338713926613282547280, 7.70768377695731666764039625399, 8.998706342937492416711074710729, 9.455916773834524384296350446944, 10.14481938073718668584361223014

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