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Properties

Label 2-1183-91.90-c0-0-0
Degree $2$
Conductor $1183$
Sign $-0.969 + 0.246i$
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  + 1.80i·2-s − 2.24·4-s + i·7-s − 2.24i·8-s + 9-s + 1.24i·11-s − 1.80·14-s + 1.80·16-s + 1.80i·18-s − 2.24·22-s + 0.445·23-s − 25-s − 2.24i·28-s − 1.80·29-s + 1.00i·32-s + ⋯
L(s)  = 1  + 1.80i·2-s − 2.24·4-s + i·7-s − 2.24i·8-s + 9-s + 1.24i·11-s − 1.80·14-s + 1.80·16-s + 1.80i·18-s − 2.24·22-s + 0.445·23-s − 25-s − 2.24i·28-s − 1.80·29-s + 1.00i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.969 + 0.246i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9023761200\)
\(L(\frac12)\) \(\approx\) \(0.9023761200\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 - 1.80iT - T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 - 1.24iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 0.445T + T^{2} \)
29 \( 1 + 1.80T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 0.445iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.80T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 0.445iT - T^{2} \)
71 \( 1 - 0.445iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.770706170910038950064486010009, −9.461059037969417374775953028684, −8.634078634667953731023776926412, −7.52716211765282402974896214580, −7.32556267748464310535553215214, −6.26320180209223436165339317113, −5.53869468351541031051502742697, −4.71273680041341470861909838631, −3.89846220124119449415311332258, −2.02837111043016220210040011736, 0.862236601067180965211897383137, 1.96981164823909590900834375002, 3.34972460621826913744975681449, 3.89853492472962480270853992766, 4.75590395667154621390750459400, 5.94472746060503504294586760485, 7.26127268510077804611076655077, 8.082621887487405995547355150940, 9.189861649089819073575774516650, 9.688693222320426345901218153024

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