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  + 0.630·2-s − 1.60·4-s + 3.89·5-s − 3.68·7-s − 2.27·8-s + 2.45·10-s + 4.96·11-s − 1.69·13-s − 2.32·14-s + 1.77·16-s − 5.52·17-s + 4.21·19-s − 6.24·20-s + 3.13·22-s + 2.77·23-s + 10.1·25-s − 1.06·26-s + 5.90·28-s + 2.31·29-s − 0.199·31-s + 5.66·32-s − 3.48·34-s − 14.3·35-s + 1.79·37-s + 2.65·38-s − 8.85·40-s + 12.1·41-s + ⋯
L(s)  = 1  + 0.445·2-s − 0.801·4-s + 1.74·5-s − 1.39·7-s − 0.803·8-s + 0.777·10-s + 1.49·11-s − 0.468·13-s − 0.620·14-s + 0.442·16-s − 1.33·17-s + 0.966·19-s − 1.39·20-s + 0.667·22-s + 0.578·23-s + 2.03·25-s − 0.209·26-s + 1.11·28-s + 0.429·29-s − 0.0358·31-s + 1.00·32-s − 0.597·34-s − 2.42·35-s + 0.294·37-s + 0.431·38-s − 1.39·40-s + 1.90·41-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

\(L(1)\) \(\approx\) \(2.204132368\)
\(L(\frac12)\) \(\approx\) \(2.204132368\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.630T + 2T^{2} \)
5 \( 1 - 3.89T + 5T^{2} \)
7 \( 1 + 3.68T + 7T^{2} \)
11 \( 1 - 4.96T + 11T^{2} \)
13 \( 1 + 1.69T + 13T^{2} \)
17 \( 1 + 5.52T + 17T^{2} \)
19 \( 1 - 4.21T + 19T^{2} \)
23 \( 1 - 2.77T + 23T^{2} \)
29 \( 1 - 2.31T + 29T^{2} \)
31 \( 1 + 0.199T + 31T^{2} \)
37 \( 1 - 1.79T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 + 5.34T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 1.32T + 53T^{2} \)
59 \( 1 + 5.78T + 59T^{2} \)
61 \( 1 + 0.0766T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 5.20T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 - 7.82T + 79T^{2} \)
83 \( 1 + 2.55T + 83T^{2} \)
89 \( 1 - 4.35T + 89T^{2} \)
97 \( 1 - 9.02T + 97T^{2} \)
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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

−9.179678985276129169805906357332, −8.850228302807517163593374155659, −7.19730313509787437844807403366, −6.35842809573470315694051390426, −6.07444175361072807422979078799, −5.12969875932958042995824432580, −4.24860611356543718345711548355, −3.25214284843653085456393215233, −2.37196706815865171227513388678, −0.940626712707947744438495944189, 0.940626712707947744438495944189, 2.37196706815865171227513388678, 3.25214284843653085456393215233, 4.24860611356543718345711548355, 5.12969875932958042995824432580, 6.07444175361072807422979078799, 6.35842809573470315694051390426, 7.19730313509787437844807403366, 8.850228302807517163593374155659, 9.179678985276129169805906357332

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