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Properties

Label 2-369600-1.1-c1-0-128
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s − 11-s + 4·13-s − 6·17-s + 6·19-s − 21-s − 8·23-s − 27-s + 33-s + 2·37-s − 4·39-s − 8·41-s + 8·43-s + 49-s + 6·51-s − 10·53-s − 6·57-s + 4·59-s + 2·61-s + 63-s + 12·67-s + 8·69-s − 8·71-s + 12·73-s − 77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 1.45·17-s + 1.37·19-s − 0.218·21-s − 1.66·23-s − 0.192·27-s + 0.174·33-s + 0.328·37-s − 0.640·39-s − 1.24·41-s + 1.21·43-s + 1/7·49-s + 0.840·51-s − 1.37·53-s − 0.794·57-s + 0.520·59-s + 0.256·61-s + 0.125·63-s + 1.46·67-s + 0.963·69-s − 0.949·71-s + 1.40·73-s − 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.113367165\)
\(L(\frac12)\) \(\approx\) \(2.113367165\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−12.42649808820567, −11.98830575687908, −11.51570950993206, −11.08289476181383, −10.96714821749249, −10.22893095536745, −9.835729758010289, −9.451645791287645, −8.693706219178329, −8.506862669694370, −7.886570038284424, −7.465680537074814, −6.952130408917957, −6.343233056685508, −6.042436375529774, −5.572691734976986, −4.959752535914518, −4.582030405833857, −3.982159163508841, −3.544892038479560, −2.931788597673474, −2.036645316597292, −1.867584813109622, −0.9678275096676332, −0.4503425342386426, 0.4503425342386426, 0.9678275096676332, 1.867584813109622, 2.036645316597292, 2.931788597673474, 3.544892038479560, 3.982159163508841, 4.582030405833857, 4.959752535914518, 5.572691734976986, 6.042436375529774, 6.343233056685508, 6.952130408917957, 7.465680537074814, 7.886570038284424, 8.506862669694370, 8.693706219178329, 9.451645791287645, 9.835729758010289, 10.22893095536745, 10.96714821749249, 11.08289476181383, 11.51570950993206, 11.98830575687908, 12.42649808820567

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