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Properties

Label 2-1183-91.55-c0-0-1
Degree $2$
Conductor $1183$
Sign $-0.990 + 0.134i$
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  + (0.900 + 1.56i)2-s + (−1.12 + 1.94i)4-s + (−0.5 + 0.866i)7-s − 2.24·8-s + (−0.5 + 0.866i)9-s + (−0.623 − 1.07i)11-s − 1.80·14-s + (−0.900 − 1.56i)16-s − 1.80·18-s + (1.12 − 1.94i)22-s + (0.222 + 0.385i)23-s + 25-s + (−1.12 − 1.94i)28-s + (0.900 + 1.56i)29-s + (0.500 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.900 + 1.56i)2-s + (−1.12 + 1.94i)4-s + (−0.5 + 0.866i)7-s − 2.24·8-s + (−0.5 + 0.866i)9-s + (−0.623 − 1.07i)11-s − 1.80·14-s + (−0.900 − 1.56i)16-s − 1.80·18-s + (1.12 − 1.94i)22-s + (0.222 + 0.385i)23-s + 25-s + (−1.12 − 1.94i)28-s + (0.900 + 1.56i)29-s + (0.500 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.990 + 0.134i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.990 + 0.134i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.298672906\)
\(L(\frac12)\) \(\approx\) \(1.298672906\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
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

−10.41761836920726529795173960951, −8.963354404604025300689209724696, −8.597334148911916402769705130547, −7.82004526825595375825020116107, −6.90924790808649614508013727993, −6.09461646564550951711612624837, −5.36917023065666130134037465954, −4.91082963089139420612313416087, −3.46731441658396192930491545863, −2.71700544961423916570107770985, 0.882784683440378454382926322735, 2.42115420017392378866039383174, 3.18865786731047742011701616473, 4.20389513877995572958941862345, 4.77977803047232652221772249196, 5.96173431760018756325791766710, 6.79750001443659394551372584849, 7.974792559886379703040978817608, 9.294058800379062299129207235083, 9.775518069108438058361886926533

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