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Properties

Label 2-369600-1.1-c1-0-82
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 − 6·13-s + 7·17-s + 5·19-s − 21-s + 23-s + 27-s + 5·29-s − 8·31-s − 33-s − 2·37-s − 6·39-s + 12·41-s − 11·43-s − 8·47-s + 49-s + 7·51-s − 11·53-s + 5·57-s + 5·59-s − 7·61-s − 63-s − 2·67-s + 69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 1.69·17-s + 1.14·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 0.928·29-s − 1.43·31-s − 0.174·33-s − 0.328·37-s − 0.960·39-s + 1.87·41-s − 1.67·43-s − 1.16·47-s + 1/7·49-s + 0.980·51-s − 1.51·53-s + 0.662·57-s + 0.650·59-s − 0.896·61-s − 0.125·63-s − 0.244·67-s + 0.120·69-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

\(L(1)\) \(\approx\) \(2.097711550\)
\(L(\frac12)\) \(\approx\) \(2.097711550\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−12.55288867390874, −12.07860429672488, −11.80714477391643, −11.10470098438029, −10.61961708163846, −10.04601073738302, −9.661805664610268, −9.563929283247534, −8.984290895510486, −8.217667960525244, −7.938481376332968, −7.427284726198312, −7.189445591534587, −6.599957660660689, −5.933422471471162, −5.408911600112916, −4.988721527700073, −4.629335391598211, −3.782414466246946, −3.291949034928340, −2.973098603114373, −2.455153594297835, −1.719797083812183, −1.175767332979478, −0.3647561863329844, 0.3647561863329844, 1.175767332979478, 1.719797083812183, 2.455153594297835, 2.973098603114373, 3.291949034928340, 3.782414466246946, 4.629335391598211, 4.988721527700073, 5.408911600112916, 5.933422471471162, 6.599957660660689, 7.189445591534587, 7.427284726198312, 7.938481376332968, 8.217667960525244, 8.984290895510486, 9.563929283247534, 9.661805664610268, 10.04601073738302, 10.61961708163846, 11.10470098438029, 11.80714477391643, 12.07860429672488, 12.55288867390874

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