Properties

Label 2-369600-1.1-c1-0-133
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 + 2·13-s + 6·17-s − 4·19-s + 21-s − 8·23-s + 27-s + 6·29-s − 4·31-s − 33-s + 2·37-s + 2·39-s − 6·41-s − 12·43-s + 49-s + 6·51-s + 2·53-s − 4·57-s + 4·59-s + 2·61-s + 63-s + 8·67-s − 8·69-s + 8·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.218·21-s − 1.66·23-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 1.82·43-s + 1/7·49-s + 0.840·51-s + 0.274·53-s − 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.125·63-s + 0.977·67-s − 0.963·69-s + 0.949·71-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: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.253314112\)
\(L(\frac12)\) \(\approx\) \(3.253314112\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.52478261927379, −11.97848839402374, −11.72310158117119, −11.20287252105763, −10.44316445355699, −10.23696094358825, −9.959087420704282, −9.303390387093813, −8.726896663222409, −8.285157500875865, −8.090569254714627, −7.599808848632676, −7.041775301081632, −6.360724213657699, −6.169878333720094, −5.385404484087363, −5.051582791893604, −4.453842681225111, −3.736945135768605, −3.595287061255275, −2.919737249079896, −2.191512062056003, −1.850895255815996, −1.185314864905352, −0.4527356348880673, 0.4527356348880673, 1.185314864905352, 1.850895255815996, 2.191512062056003, 2.919737249079896, 3.595287061255275, 3.736945135768605, 4.453842681225111, 5.051582791893604, 5.385404484087363, 6.169878333720094, 6.360724213657699, 7.041775301081632, 7.599808848632676, 8.090569254714627, 8.285157500875865, 8.726896663222409, 9.303390387093813, 9.959087420704282, 10.23696094358825, 10.44316445355699, 11.20287252105763, 11.72310158117119, 11.97848839402374, 12.52478261927379

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