Properties

Degree $2$
Conductor $331200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s − 2·13-s − 6·17-s − 4·19-s − 23-s − 2·29-s − 2·37-s − 10·41-s − 4·43-s − 7·49-s − 6·53-s − 4·59-s + 10·61-s − 12·67-s + 8·71-s − 10·73-s − 8·79-s + 4·83-s − 18·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s − 0.208·23-s − 0.371·29-s − 0.328·37-s − 1.56·41-s − 0.609·43-s − 49-s − 0.824·53-s − 0.520·59-s + 1.28·61-s − 1.46·67-s + 0.949·71-s − 1.17·73-s − 0.900·79-s + 0.439·83-s − 1.90·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(331200\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{331200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 331200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3825486075\)
\(L(\frac12)\) \(\approx\) \(0.3825486075\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
23 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.57996783700937, −12.07197997558131, −11.74033377136356, −11.14613075400211, −10.99252217404713, −10.22679806674383, −9.882757596082566, −9.401506125385263, −8.885017099811664, −8.502654201546117, −8.185006718417264, −7.365006147987980, −6.918510627862814, −6.650061938445591, −6.125985605003468, −5.638668630291016, −4.858658385922290, −4.521359011050093, −4.127347567322694, −3.418639206820075, −3.029365985516445, −2.066356674604116, −1.919871149529107, −1.199006830100236, −0.1585725193452861, 0.1585725193452861, 1.199006830100236, 1.919871149529107, 2.066356674604116, 3.029365985516445, 3.418639206820075, 4.127347567322694, 4.521359011050093, 4.858658385922290, 5.638668630291016, 6.125985605003468, 6.650061938445591, 6.918510627862814, 7.365006147987980, 8.185006718417264, 8.502654201546117, 8.885017099811664, 9.401506125385263, 9.882757596082566, 10.22679806674383, 10.99252217404713, 11.14613075400211, 11.74033377136356, 12.07197997558131, 12.57996783700937

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