Properties

Label 2-1183-91.69-c0-0-5
Degree $2$
Conductor $1183$
Sign $-0.499 + 0.866i$
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  + (0.385 + 0.222i)2-s + (−0.400 − 0.694i)4-s + (−0.866 + 0.5i)7-s − 0.801i·8-s + (−0.5 − 0.866i)9-s + (−1.56 − 0.900i)11-s − 0.445·14-s + (−0.222 + 0.385i)16-s − 0.445i·18-s + (−0.400 − 0.694i)22-s + (0.623 − 1.07i)23-s − 25-s + (0.694 + 0.400i)28-s + (0.222 − 0.385i)29-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.385 + 0.222i)2-s + (−0.400 − 0.694i)4-s + (−0.866 + 0.5i)7-s − 0.801i·8-s + (−0.5 − 0.866i)9-s + (−1.56 − 0.900i)11-s − 0.445·14-s + (−0.222 + 0.385i)16-s − 0.445i·18-s + (−0.400 − 0.694i)22-s + (0.623 − 1.07i)23-s − 25-s + (0.694 + 0.400i)28-s + (0.222 − 0.385i)29-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.499 + 0.866i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.499 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6286429199\)
\(L(\frac12)\) \(\approx\) \(0.6286429199\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 \)
good2 \( 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + (1.56 + 0.900i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (1.07 + 0.623i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.07 + 0.623i)T + (0.5 - 0.866i)T^{2} \)
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

−9.738260222229850093450566364253, −8.892930608049712692590132223203, −8.233936241205362841505167445621, −6.95277257310320788845356982190, −5.99041009311565771818719593870, −5.74059673786470362816093837827, −4.64829770245183925765025959500, −3.45770188230477458205202466461, −2.59326815834271383735362469095, −0.46246844166766787470249264948, 2.31773025899212567698055362782, 3.08124370031055980767532441767, 4.15291728925564051343313597353, 5.04925188764025914687450419383, 5.77245720157717041110395799796, 7.29214038976976874643878621614, 7.62259513965049867799151388154, 8.510619940444083116350698702591, 9.530877922004027921744150123099, 10.23242928589298149343564125924

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