Properties

Label 2-369600-1.1-c1-0-105
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 − 4·17-s − 2·19-s + 21-s − 2·23-s − 27-s + 6·29-s − 6·31-s + 33-s + 6·37-s + 4·39-s − 8·41-s − 43-s + 9·47-s + 49-s + 4·51-s + 13·53-s + 2·57-s + 10·59-s + 10·61-s − 63-s + 4·67-s + 2·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.970·17-s − 0.458·19-s + 0.218·21-s − 0.417·23-s − 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.174·33-s + 0.986·37-s + 0.640·39-s − 1.24·41-s − 0.152·43-s + 1.31·47-s + 1/7·49-s + 0.560·51-s + 1.78·53-s + 0.264·57-s + 1.30·59-s + 1.28·61-s − 0.125·63-s + 0.488·67-s + 0.240·69-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

\(L(1)\) \(\approx\) \(1.383666548\)
\(L(\frac12)\) \(\approx\) \(1.383666548\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−12.57377066268231, −11.87484316034732, −11.73513696983882, −11.23475659792320, −10.52273467528453, −10.32030747855046, −9.955344752022614, −9.331128358951171, −8.925924087692098, −8.393923276615511, −7.940994050366679, −7.241986259220656, −6.992901601785737, −6.525933447323653, −6.040919028952056, −5.322137576143591, −5.204875769430029, −4.455914239857413, −4.070843550928177, −3.552674713638999, −2.696622084311266, −2.322807854776017, −1.875360212836094, −0.8217193923765733, −0.4023954479969369, 0.4023954479969369, 0.8217193923765733, 1.875360212836094, 2.322807854776017, 2.696622084311266, 3.552674713638999, 4.070843550928177, 4.455914239857413, 5.204875769430029, 5.322137576143591, 6.040919028952056, 6.525933447323653, 6.992901601785737, 7.241986259220656, 7.940994050366679, 8.393923276615511, 8.925924087692098, 9.331128358951171, 9.955344752022614, 10.32030747855046, 10.52273467528453, 11.23475659792320, 11.73513696983882, 11.87484316034732, 12.57377066268231

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