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  + 1.35·2-s − 0.160·4-s − 2.74·5-s − 0.283·7-s − 2.93·8-s − 3.71·10-s + 4.12·11-s + 0.0271·13-s − 0.384·14-s − 3.65·16-s + 1.28·17-s − 5.72·19-s + 0.440·20-s + 5.59·22-s + 5.97·23-s + 2.51·25-s + 0.0368·26-s + 0.0455·28-s + 2.55·29-s − 2.02·31-s + 0.905·32-s + 1.74·34-s + 0.777·35-s + 2.42·37-s − 7.76·38-s + 8.03·40-s + 11.0·41-s + ⋯
L(s)  = 1  + 0.959·2-s − 0.0802·4-s − 1.22·5-s − 0.107·7-s − 1.03·8-s − 1.17·10-s + 1.24·11-s + 0.00754·13-s − 0.102·14-s − 0.913·16-s + 0.312·17-s − 1.31·19-s + 0.0984·20-s + 1.19·22-s + 1.24·23-s + 0.503·25-s + 0.00723·26-s + 0.00860·28-s + 0.474·29-s − 0.364·31-s + 0.160·32-s + 0.299·34-s + 0.131·35-s + 0.399·37-s − 1.26·38-s + 1.27·40-s + 1.72·41-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

\(L(1)\) \(\approx\) \(1.850353682\)
\(L(\frac12)\) \(\approx\) \(1.850353682\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.35T + 2T^{2} \)
5 \( 1 + 2.74T + 5T^{2} \)
7 \( 1 + 0.283T + 7T^{2} \)
11 \( 1 - 4.12T + 11T^{2} \)
13 \( 1 - 0.0271T + 13T^{2} \)
17 \( 1 - 1.28T + 17T^{2} \)
19 \( 1 + 5.72T + 19T^{2} \)
23 \( 1 - 5.97T + 23T^{2} \)
29 \( 1 - 2.55T + 29T^{2} \)
31 \( 1 + 2.02T + 31T^{2} \)
37 \( 1 - 2.42T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + 4.54T + 47T^{2} \)
53 \( 1 - 9.30T + 53T^{2} \)
59 \( 1 - 9.94T + 59T^{2} \)
61 \( 1 - 8.17T + 61T^{2} \)
67 \( 1 - 4.40T + 67T^{2} \)
71 \( 1 - 3.80T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 - 6.69T + 79T^{2} \)
83 \( 1 - 4.32T + 83T^{2} \)
89 \( 1 + 0.746T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−8.936915464536973320181212876415, −8.391200308786206227372542552620, −7.38174040045307799174717263422, −6.61307967051733956806723981411, −5.86564263801886190400259397354, −4.80283107666351012706813400405, −4.08514881967396252029764307237, −3.67773855518284955872201957363, −2.58525179049367731892609542942, −0.77466463474000457330126862464, 0.77466463474000457330126862464, 2.58525179049367731892609542942, 3.67773855518284955872201957363, 4.08514881967396252029764307237, 4.80283107666351012706813400405, 5.86564263801886190400259397354, 6.61307967051733956806723981411, 7.38174040045307799174717263422, 8.391200308786206227372542552620, 8.936915464536973320181212876415

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