Properties

Degree $2$
Conductor $2169$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.60·2-s + 4.77·4-s + 1.69·5-s − 1.30·7-s + 7.23·8-s + 4.40·10-s + 3.27·11-s + 4.30·13-s − 3.39·14-s + 9.27·16-s + 1.02·17-s − 7.01·19-s + 8.08·20-s + 8.51·22-s − 0.835·23-s − 2.13·25-s + 11.2·26-s − 6.24·28-s + 1.11·29-s − 3.97·31-s + 9.69·32-s + 2.66·34-s − 2.20·35-s − 11.3·37-s − 18.2·38-s + 12.2·40-s − 1.22·41-s + ⋯
L(s)  = 1  + 1.84·2-s + 2.38·4-s + 0.756·5-s − 0.493·7-s + 2.55·8-s + 1.39·10-s + 0.986·11-s + 1.19·13-s − 0.908·14-s + 2.31·16-s + 0.248·17-s − 1.60·19-s + 1.80·20-s + 1.81·22-s − 0.174·23-s − 0.427·25-s + 2.19·26-s − 1.17·28-s + 0.207·29-s − 0.713·31-s + 1.71·32-s + 0.457·34-s − 0.373·35-s − 1.85·37-s − 2.96·38-s + 1.93·40-s − 0.191·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2169\)    =    \(3^{2} \cdot 241\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2169} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2169,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.539429821\)
\(L(\frac12)\) \(\approx\) \(6.539429821\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
241 \( 1 + T \)
good2 \( 1 - 2.60T + 2T^{2} \)
5 \( 1 - 1.69T + 5T^{2} \)
7 \( 1 + 1.30T + 7T^{2} \)
11 \( 1 - 3.27T + 11T^{2} \)
13 \( 1 - 4.30T + 13T^{2} \)
17 \( 1 - 1.02T + 17T^{2} \)
19 \( 1 + 7.01T + 19T^{2} \)
23 \( 1 + 0.835T + 23T^{2} \)
29 \( 1 - 1.11T + 29T^{2} \)
31 \( 1 + 3.97T + 31T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + 1.22T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 0.151T + 47T^{2} \)
53 \( 1 + 3.02T + 53T^{2} \)
59 \( 1 - 4.15T + 59T^{2} \)
61 \( 1 - 5.62T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + 1.66T + 79T^{2} \)
83 \( 1 - 2.34T + 83T^{2} \)
89 \( 1 + 18.1T + 89T^{2} \)
97 \( 1 + 7.17T + 97T^{2} \)
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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

−9.097029595863094145306457293580, −8.201601297625781184960422866737, −6.88183129597797247590962712760, −6.47666022395559537533276920982, −5.87566841590927638085770105298, −5.14937152329153010746102216464, −3.91994584278796338296392549665, −3.71598413957196087374871844966, −2.42690416047087321563416459680, −1.58422942899643427705506895548, 1.58422942899643427705506895548, 2.42690416047087321563416459680, 3.71598413957196087374871844966, 3.91994584278796338296392549665, 5.14937152329153010746102216464, 5.87566841590927638085770105298, 6.47666022395559537533276920982, 6.88183129597797247590962712760, 8.201601297625781184960422866737, 9.097029595863094145306457293580

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