Properties

Degree $2$
Conductor $2352$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 9-s + 6·11-s − 3·13-s + 2·15-s + 4·17-s + 5·19-s + 4·23-s − 25-s + 27-s − 4·29-s − 7·31-s + 6·33-s − 9·37-s − 3·39-s − 2·41-s + 43-s + 2·45-s − 2·47-s + 4·51-s + 8·53-s + 12·55-s + 5·57-s + 10·61-s − 6·65-s + 15·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.80·11-s − 0.832·13-s + 0.516·15-s + 0.970·17-s + 1.14·19-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.25·31-s + 1.04·33-s − 1.47·37-s − 0.480·39-s − 0.312·41-s + 0.152·43-s + 0.298·45-s − 0.291·47-s + 0.560·51-s + 1.09·53-s + 1.61·55-s + 0.662·57-s + 1.28·61-s − 0.744·65-s + 1.83·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2352} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.011684828\)
\(L(\frac12)\) \(\approx\) \(3.011684828\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−19.19515002228155, −18.49973686156358, −17.75403101896590, −17.02797437350110, −16.80524040725248, −15.87568249763733, −14.95673361472606, −14.39252275759059, −14.11295134345918, −13.33510900206416, −12.54248300398600, −11.90574851703474, −11.22022682160535, −10.11063836462406, −9.590661490266592, −9.188194446029912, −8.362327019091454, −7.200950541943149, −6.945698047620078, −5.729620730356640, −5.194220285082706, −3.924568440756354, −3.270353027890634, −2.055914308343460, −1.227709158501730, 1.227709158501730, 2.055914308343460, 3.270353027890634, 3.924568440756354, 5.194220285082706, 5.729620730356640, 6.945698047620078, 7.200950541943149, 8.362327019091454, 9.188194446029912, 9.590661490266592, 10.11063836462406, 11.22022682160535, 11.90574851703474, 12.54248300398600, 13.33510900206416, 14.11295134345918, 14.39252275759059, 14.95673361472606, 15.87568249763733, 16.80524040725248, 17.02797437350110, 17.75403101896590, 18.49973686156358, 19.19515002228155

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