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Properties

Label 2-1183-91.48-c0-0-0
Degree $2$
Conductor $1183$
Sign $-0.434 - 0.900i$
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  + (−0.222 + 0.385i)2-s + (0.400 + 0.694i)4-s + (0.5 + 0.866i)7-s − 0.801·8-s + (−0.5 − 0.866i)9-s + (−0.900 + 1.56i)11-s − 0.445·14-s + (−0.222 + 0.385i)16-s + 0.445·18-s + (−0.400 − 0.694i)22-s + (−0.623 + 1.07i)23-s + 25-s + (−0.400 + 0.694i)28-s + (0.222 − 0.385i)29-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.222 + 0.385i)2-s + (0.400 + 0.694i)4-s + (0.5 + 0.866i)7-s − 0.801·8-s + (−0.5 − 0.866i)9-s + (−0.900 + 1.56i)11-s − 0.445·14-s + (−0.222 + 0.385i)16-s + 0.445·18-s + (−0.400 − 0.694i)22-s + (−0.623 + 1.07i)23-s + 25-s + (−0.400 + 0.694i)28-s + (0.222 − 0.385i)29-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.434 - 0.900i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.434 - 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9099040356\)
\(L(\frac12)\) \(\approx\) \(0.9099040356\)
\(L(1)\) 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 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.24T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
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

−10.02037374967814772728186702550, −9.249893704388859111564724767712, −8.532719879363052679601800590020, −7.72846129997180058963926680096, −7.10161776426868740534893720454, −6.10201391364781150502207267925, −5.30147595927671952609110566202, −4.15940520495500395914522996128, −2.92830832619068856053975567120, −2.10368287189603853147562062508, 0.850423680642288524398451360850, 2.31835832649821153251033532971, 3.19269974087047810285028504191, 4.66020984733527627161796058061, 5.46588898682557444597271943541, 6.26350792786030402859017122412, 7.28261721175466277696094932745, 8.290304038956141347102315122331, 8.698748523919830342216948424974, 10.13270095300552545271087459654

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