Properties

Degree $2$
Conductor $29400$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 5·11-s + 2·13-s − 6·17-s − 2·19-s − 5·23-s − 27-s − 5·29-s − 4·31-s + 5·33-s − 37-s − 2·39-s − 12·41-s − 5·43-s − 2·47-s + 6·51-s − 14·53-s + 2·57-s + 2·59-s + 5·67-s + 5·69-s − 9·71-s − 10·73-s + 11·79-s + 81-s − 16·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.554·13-s − 1.45·17-s − 0.458·19-s − 1.04·23-s − 0.192·27-s − 0.928·29-s − 0.718·31-s + 0.870·33-s − 0.164·37-s − 0.320·39-s − 1.87·41-s − 0.762·43-s − 0.291·47-s + 0.840·51-s − 1.92·53-s + 0.264·57-s + 0.260·59-s + 0.610·67-s + 0.601·69-s − 1.06·71-s − 1.17·73-s + 1.23·79-s + 1/9·81-s − 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29400\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{29400} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 29400,\ (\ :1/2),\ 1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.70702333654038, −15.33931338617329, −14.76540046961425, −13.96629569506237, −13.44124163391507, −12.99101449166774, −12.69822774212235, −11.87235211476121, −11.30217274654765, −10.95176460428579, −10.37184957477885, −9.930854431675511, −9.222714704509438, −8.428679285118434, −8.203420298230281, −7.352195395345940, −6.860843330481127, −6.168725944763822, −5.683458807233513, −5.001874402977524, −4.509970524751258, −3.743111490661461, −3.019338151971182, −2.092485558037704, −1.624443240525449, 0, 0, 1.624443240525449, 2.092485558037704, 3.019338151971182, 3.743111490661461, 4.509970524751258, 5.001874402977524, 5.683458807233513, 6.168725944763822, 6.860843330481127, 7.352195395345940, 8.203420298230281, 8.428679285118434, 9.222714704509438, 9.930854431675511, 10.37184957477885, 10.95176460428579, 11.30217274654765, 11.87235211476121, 12.69822774212235, 12.99101449166774, 13.44124163391507, 13.96629569506237, 14.76540046961425, 15.33931338617329, 15.70702333654038

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