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  − 0.487·2-s − 1.76·4-s − 0.961·5-s − 4.61·7-s + 1.83·8-s + 0.468·10-s − 1.93·11-s − 3.85·13-s + 2.24·14-s + 2.63·16-s − 5.40·17-s − 4.17·19-s + 1.69·20-s + 0.945·22-s + 1.42·23-s − 4.07·25-s + 1.87·26-s + 8.13·28-s + 4.85·29-s − 7.24·31-s − 4.94·32-s + 2.63·34-s + 4.43·35-s + 7.12·37-s + 2.03·38-s − 1.76·40-s − 9.18·41-s + ⋯
L(s)  = 1  − 0.344·2-s − 0.881·4-s − 0.430·5-s − 1.74·7-s + 0.648·8-s + 0.148·10-s − 0.584·11-s − 1.06·13-s + 0.600·14-s + 0.657·16-s − 1.31·17-s − 0.956·19-s + 0.379·20-s + 0.201·22-s + 0.297·23-s − 0.814·25-s + 0.368·26-s + 1.53·28-s + 0.902·29-s − 1.30·31-s − 0.874·32-s + 0.451·34-s + 0.750·35-s + 1.17·37-s + 0.329·38-s − 0.278·40-s − 1.43·41-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

\(L(1)\) \(\approx\) \(0.1687707048\)
\(L(\frac12)\) \(\approx\) \(0.1687707048\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.487T + 2T^{2} \)
5 \( 1 + 0.961T + 5T^{2} \)
7 \( 1 + 4.61T + 7T^{2} \)
11 \( 1 + 1.93T + 11T^{2} \)
13 \( 1 + 3.85T + 13T^{2} \)
17 \( 1 + 5.40T + 17T^{2} \)
19 \( 1 + 4.17T + 19T^{2} \)
23 \( 1 - 1.42T + 23T^{2} \)
29 \( 1 - 4.85T + 29T^{2} \)
31 \( 1 + 7.24T + 31T^{2} \)
37 \( 1 - 7.12T + 37T^{2} \)
41 \( 1 + 9.18T + 41T^{2} \)
43 \( 1 - 2.93T + 43T^{2} \)
47 \( 1 + 2.48T + 47T^{2} \)
53 \( 1 + 5.64T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 + 7.30T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 - 0.240T + 73T^{2} \)
79 \( 1 + 3.15T + 79T^{2} \)
83 \( 1 - 2.46T + 83T^{2} \)
89 \( 1 + 5.55T + 89T^{2} \)
97 \( 1 - 5.27T + 97T^{2} \)
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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

−9.151494739712917038151514544125, −8.437480972683827551967985663026, −7.56409769858103038280669204435, −6.81827135574788534505425038011, −6.01974381030064924698817360366, −4.93318223424281728457917349405, −4.19444718650555426715272080747, −3.30910714656706357608688161764, −2.27530469017381419724223355674, −0.26474122094783773564724981735, 0.26474122094783773564724981735, 2.27530469017381419724223355674, 3.30910714656706357608688161764, 4.19444718650555426715272080747, 4.93318223424281728457917349405, 6.01974381030064924698817360366, 6.81827135574788534505425038011, 7.56409769858103038280669204435, 8.437480972683827551967985663026, 9.151494739712917038151514544125

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