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Properties

Label 2-369600-1.1-c1-0-108
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 + 3·13-s − 6·17-s − 3·19-s + 21-s + 5·23-s + 27-s + 7·29-s − 3·31-s − 33-s − 4·37-s + 3·39-s − 8·41-s − 13·43-s + 49-s − 6·51-s + 12·53-s − 3·57-s + 8·59-s + 10·61-s + 63-s − 12·67-s + 5·69-s + 5·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 1.45·17-s − 0.688·19-s + 0.218·21-s + 1.04·23-s + 0.192·27-s + 1.29·29-s − 0.538·31-s − 0.174·33-s − 0.657·37-s + 0.480·39-s − 1.24·41-s − 1.98·43-s + 1/7·49-s − 0.840·51-s + 1.64·53-s − 0.397·57-s + 1.04·59-s + 1.28·61-s + 0.125·63-s − 1.46·67-s + 0.601·69-s + 0.593·71-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.734739333\)
\(L(\frac12)\) \(\approx\) \(2.734739333\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−12.77148415300076, −11.85324393219290, −11.63776595143909, −11.19473947913193, −10.55155335671135, −10.33579968366364, −9.892713210923468, −9.116729817769469, −8.661172076602477, −8.510628160104562, −8.241314509450333, −7.292866010546101, −7.027115540905758, −6.596962137612085, −6.120743239333210, −5.335634840622376, −5.001956595442576, −4.444779775007811, −3.970122116153390, −3.403410694354634, −2.894382366995464, −2.263804109527252, −1.818562180854437, −1.204585606588359, −0.4086944093152324, 0.4086944093152324, 1.204585606588359, 1.818562180854437, 2.263804109527252, 2.894382366995464, 3.403410694354634, 3.970122116153390, 4.444779775007811, 5.001956595442576, 5.335634840622376, 6.120743239333210, 6.596962137612085, 7.027115540905758, 7.292866010546101, 8.241314509450333, 8.510628160104562, 8.661172076602477, 9.116729817769469, 9.892713210923468, 10.33579968366364, 10.55155335671135, 11.19473947913193, 11.63776595143909, 11.85324393219290, 12.77148415300076

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