Properties

Label 2-369600-1.1-c1-0-113
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 − 4·19-s + 21-s − 8·23-s − 27-s − 2·29-s − 4·31-s − 33-s + 2·37-s − 2·39-s + 10·41-s − 4·43-s + 8·47-s + 49-s − 6·51-s + 10·53-s + 4·57-s + 4·59-s + 10·61-s − 63-s + 8·69-s − 8·71-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 − 0.917·19-s + 0.218·21-s − 1.66·23-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.174·33-s + 0.328·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.840·51-s + 1.37·53-s + 0.529·57-s + 0.520·59-s + 1.28·61-s − 0.125·63-s + 0.963·69-s − 0.949·71-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: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.841973858\)
\(L(\frac12)\) \(\approx\) \(1.841973858\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.40988358787530, −12.07417498633249, −11.64409801364863, −11.21245943203319, −10.51801777206253, −10.40865371476638, −9.864295580873586, −9.338718721466425, −8.993006283971392, −8.290073826303489, −7.916808884395699, −7.491515204158108, −6.852011266491535, −6.454008913189203, −5.940772431233222, −5.526141943642593, −5.281089948856194, −4.231592417858205, −4.005118273410482, −3.686312390353985, −2.842965264844857, −2.278764764357326, −1.667530169223928, −0.9932505133006185, −0.4190202364084392, 0.4190202364084392, 0.9932505133006185, 1.667530169223928, 2.278764764357326, 2.842965264844857, 3.686312390353985, 4.005118273410482, 4.231592417858205, 5.281089948856194, 5.526141943642593, 5.940772431233222, 6.454008913189203, 6.852011266491535, 7.491515204158108, 7.916808884395699, 8.290073826303489, 8.993006283971392, 9.338718721466425, 9.864295580873586, 10.40865371476638, 10.51801777206253, 11.21245943203319, 11.64409801364863, 12.07417498633249, 12.40988358787530

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