Properties

Degree $2$
Conductor $5166$
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-s + 4-s + 5-s + 7-s − 8-s − 10-s + 2·11-s − 14-s + 16-s + 3·17-s − 8·19-s + 20-s − 2·22-s + 4·23-s − 4·25-s + 28-s + 5·29-s − 3·31-s − 32-s − 3·34-s + 35-s + 10·37-s + 8·38-s − 40-s + 41-s − 5·43-s + 2·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.83·19-s + 0.223·20-s − 0.426·22-s + 0.834·23-s − 4/5·25-s + 0.188·28-s + 0.928·29-s − 0.538·31-s − 0.176·32-s − 0.514·34-s + 0.169·35-s + 1.64·37-s + 1.29·38-s − 0.158·40-s + 0.156·41-s − 0.762·43-s + 0.301·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5166\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 41\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{5166} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5166,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.645107593\)
\(L(\frac12)\) \(\approx\) \(1.645107593\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 - T \)
41 \( 1 - T \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−8.367685453432787752970629971919, −7.60244313029737587320284759588, −6.77260816609395551286192496066, −6.22521992764729152029152239897, −5.45821848324230659180784215026, −4.50956591206483164540917042476, −3.68629641490415248691631929400, −2.56209811383506646800247550920, −1.81335090619490852147633806567, −0.792448834555670241545210570180, 0.792448834555670241545210570180, 1.81335090619490852147633806567, 2.56209811383506646800247550920, 3.68629641490415248691631929400, 4.50956591206483164540917042476, 5.45821848324230659180784215026, 6.22521992764729152029152239897, 6.77260816609395551286192496066, 7.60244313029737587320284759588, 8.367685453432787752970629971919

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