Properties

Label 2-369600-1.1-c1-0-0
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 + 2·19-s + 21-s + 27-s − 8·29-s − 8·31-s − 33-s + 6·37-s − 2·39-s − 8·41-s − 8·43-s − 10·47-s + 49-s − 6·51-s + 2·53-s + 2·57-s + 8·59-s − 10·61-s + 63-s − 2·67-s − 10·73-s − 77-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 + 0.458·19-s + 0.218·21-s + 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.174·33-s + 0.986·37-s − 0.320·39-s − 1.24·41-s − 1.21·43-s − 1.45·47-s + 1/7·49-s − 0.840·51-s + 0.274·53-s + 0.264·57-s + 1.04·59-s − 1.28·61-s + 0.125·63-s − 0.244·67-s − 1.17·73-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: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2388940775\)
\(L(\frac12)\) \(\approx\) \(0.2388940775\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.62745097778100, −11.94161819201824, −11.60400538365589, −11.15852871384783, −10.73009319220344, −10.22573128006878, −9.656357744527172, −9.351998816426083, −8.854799407320382, −8.429200032854577, −7.941227475806608, −7.424506794243316, −7.128227748387764, −6.575873568471250, −6.028704884580249, −5.357793791745721, −4.984231853267456, −4.528200586970405, −3.880452493745993, −3.472378191899708, −2.843588931079495, −2.261314200207264, −1.805438977601864, −1.319732835305234, −0.1120321295756546, 0.1120321295756546, 1.319732835305234, 1.805438977601864, 2.261314200207264, 2.843588931079495, 3.472378191899708, 3.880452493745993, 4.528200586970405, 4.984231853267456, 5.357793791745721, 6.028704884580249, 6.575873568471250, 7.128227748387764, 7.424506794243316, 7.941227475806608, 8.429200032854577, 8.854799407320382, 9.351998816426083, 9.656357744527172, 10.22573128006878, 10.73009319220344, 11.15852871384783, 11.60400538365589, 11.94161819201824, 12.62745097778100

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