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Properties

Label 2-1183-7.2-c1-0-0
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: $\chi_{1183} (170, \cdot )$
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

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

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