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  − 0.277·2-s − 1.92·4-s + 1.23·5-s + 1.36·7-s + 1.08·8-s − 0.342·10-s + 4.69·11-s − 0.0431·13-s − 0.379·14-s + 3.54·16-s + 7.31·17-s − 0.697·19-s − 2.37·20-s − 1.30·22-s − 1.41·23-s − 3.47·25-s + 0.0119·26-s − 2.62·28-s − 8.30·29-s + 3.39·31-s − 3.16·32-s − 2.03·34-s + 1.68·35-s + 7.15·37-s + 0.193·38-s + 1.34·40-s − 5.45·41-s + ⋯
L(s)  = 1  − 0.196·2-s − 0.961·4-s + 0.551·5-s + 0.516·7-s + 0.384·8-s − 0.108·10-s + 1.41·11-s − 0.0119·13-s − 0.101·14-s + 0.885·16-s + 1.77·17-s − 0.160·19-s − 0.530·20-s − 0.278·22-s − 0.294·23-s − 0.695·25-s + 0.00235·26-s − 0.496·28-s − 1.54·29-s + 0.610·31-s − 0.558·32-s − 0.348·34-s + 0.284·35-s + 1.17·37-s + 0.0314·38-s + 0.212·40-s − 0.851·41-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

\(L(1)\) \(\approx\) \(1.717211110\)
\(L(\frac12)\) \(\approx\) \(1.717211110\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.277T + 2T^{2} \)
5 \( 1 - 1.23T + 5T^{2} \)
7 \( 1 - 1.36T + 7T^{2} \)
11 \( 1 - 4.69T + 11T^{2} \)
13 \( 1 + 0.0431T + 13T^{2} \)
17 \( 1 - 7.31T + 17T^{2} \)
19 \( 1 + 0.697T + 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 + 8.30T + 29T^{2} \)
31 \( 1 - 3.39T + 31T^{2} \)
37 \( 1 - 7.15T + 37T^{2} \)
41 \( 1 + 5.45T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 - 5.24T + 47T^{2} \)
53 \( 1 - 8.57T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + 1.86T + 71T^{2} \)
73 \( 1 - 6.47T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 2.32T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + 2.23T + 97T^{2} \)
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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

−9.124156223209943680577211949661, −8.347584247321255889439318930181, −7.74488211710293483973321371019, −6.72764736434557558865730244855, −5.72872582734408694106140789593, −5.20926034550489973422957967595, −4.08509564166718231212101500117, −3.52442687753169345961127055598, −1.89916031792756781062024569249, −0.967036410877531381680850765871, 0.967036410877531381680850765871, 1.89916031792756781062024569249, 3.52442687753169345961127055598, 4.08509564166718231212101500117, 5.20926034550489973422957967595, 5.72872582734408694106140789593, 6.72764736434557558865730244855, 7.74488211710293483973321371019, 8.347584247321255889439318930181, 9.124156223209943680577211949661

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