Properties

Label 2-369600-1.1-c1-0-115
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 + 4·19-s − 21-s − 6·23-s + 27-s − 8·29-s − 2·31-s − 33-s + 4·39-s + 2·41-s − 4·43-s + 4·47-s + 49-s + 6·51-s − 14·53-s + 4·57-s + 8·59-s − 10·61-s − 63-s − 4·67-s − 6·69-s − 4·73-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 + 0.917·19-s − 0.218·21-s − 1.25·23-s + 0.192·27-s − 1.48·29-s − 0.359·31-s − 0.174·33-s + 0.640·39-s + 0.312·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.840·51-s − 1.92·53-s + 0.529·57-s + 1.04·59-s − 1.28·61-s − 0.125·63-s − 0.488·67-s − 0.722·69-s − 0.468·73-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

\(L(1)\) \(\approx\) \(2.830099116\)
\(L(\frac12)\) \(\approx\) \(2.830099116\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.47837517995636, −12.15833142915855, −11.61097890460817, −11.12988547392936, −10.66419006586864, −10.16562433977106, −9.682064904487949, −9.433190019383484, −8.901127404106027, −8.313550793873824, −7.940273521040869, −7.483858082466225, −7.216949780014204, −6.316984842754249, −6.060899685387585, −5.526266086198006, −5.116794970257896, −4.329096823671785, −3.819000510222552, −3.339554121666554, −3.134231371817340, −2.301756372793908, −1.672558335150573, −1.242247348078039, −0.4196054770582750, 0.4196054770582750, 1.242247348078039, 1.672558335150573, 2.301756372793908, 3.134231371817340, 3.339554121666554, 3.819000510222552, 4.329096823671785, 5.116794970257896, 5.526266086198006, 6.060899685387585, 6.316984842754249, 7.216949780014204, 7.483858082466225, 7.940273521040869, 8.313550793873824, 8.901127404106027, 9.433190019383484, 9.682064904487949, 10.16562433977106, 10.66419006586864, 11.12988547392936, 11.61097890460817, 12.15833142915855, 12.47837517995636

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