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Properties

Label 2-369600-1.1-c1-0-100
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 + 2·17-s − 4·19-s + 21-s + 2·23-s + 27-s − 2·31-s − 33-s − 8·37-s + 4·39-s − 6·41-s + 4·43-s + 49-s + 2·51-s − 6·53-s − 4·57-s − 4·59-s − 6·61-s + 63-s + 4·67-s + 2·69-s + 4·73-s − 77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 0.417·23-s + 0.192·27-s − 0.359·31-s − 0.174·33-s − 1.31·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.125·63-s + 0.488·67-s + 0.240·69-s + 0.468·73-s − 0.113·77-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

\(L(1)\) \(\approx\) \(2.667828973\)
\(L(\frac12)\) \(\approx\) \(2.667828973\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
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

−12.64206401887737, −12.12959690056864, −11.51160533065045, −11.11532501960526, −10.67468694581335, −10.27547471670505, −9.862803680012704, −9.078371439137398, −8.899710729668851, −8.437992903380385, −7.970618988643170, −7.570199356770296, −7.009400368160424, −6.483963323528631, −6.084830442114141, −5.401049712873249, −5.054789047387576, −4.395759581012960, −3.908071760074793, −3.436607152241481, −2.926976937715192, −2.333542205365094, −1.577485709046841, −1.383551937926868, −0.3904536200825695, 0.3904536200825695, 1.383551937926868, 1.577485709046841, 2.333542205365094, 2.926976937715192, 3.436607152241481, 3.908071760074793, 4.395759581012960, 5.054789047387576, 5.401049712873249, 6.084830442114141, 6.483963323528631, 7.009400368160424, 7.570199356770296, 7.970618988643170, 8.437992903380385, 8.899710729668851, 9.078371439137398, 9.862803680012704, 10.27547471670505, 10.67468694581335, 11.11532501960526, 11.51160533065045, 12.12959690056864, 12.64206401887737

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