Properties

Degree $2$
Conductor $147294$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 4·11-s + 16-s − 4·17-s + 4·19-s + 2·20-s + 4·22-s + 4·23-s − 25-s + 2·29-s − 4·31-s + 32-s − 4·34-s − 12·37-s + 4·38-s + 2·40-s + 12·41-s − 8·43-s + 4·44-s + 4·46-s − 50-s − 14·53-s + 8·55-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 1.20·11-s + 1/4·16-s − 0.970·17-s + 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.685·34-s − 1.97·37-s + 0.648·38-s + 0.316·40-s + 1.87·41-s − 1.21·43-s + 0.603·44-s + 0.589·46-s − 0.141·50-s − 1.92·53-s + 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147294\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 167\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{147294} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 147294,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
167 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.74072738902597, −13.17713873205795, −12.70400755434058, −12.28070887020052, −11.60173803568472, −11.42095099294477, −10.75941725432134, −10.36320174804445, −9.588191420375259, −9.416729346263552, −8.833956906678138, −8.370432456337954, −7.548386577123415, −6.984291165994006, −6.765548701361709, −6.053874558419918, −5.742248814697675, −5.127034378250193, −4.588477792763228, −4.095681912273115, −3.363843458483034, −2.991669103838497, −2.178621156071893, −1.613823486869685, −1.182780909545540, 0, 1.182780909545540, 1.613823486869685, 2.178621156071893, 2.991669103838497, 3.363843458483034, 4.095681912273115, 4.588477792763228, 5.127034378250193, 5.742248814697675, 6.053874558419918, 6.765548701361709, 6.984291165994006, 7.548386577123415, 8.370432456337954, 8.833956906678138, 9.416729346263552, 9.588191420375259, 10.36320174804445, 10.75941725432134, 11.42095099294477, 11.60173803568472, 12.28070887020052, 12.70400755434058, 13.17713873205795, 13.74072738902597

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