Properties

Label 2-369600-1.1-c1-0-119
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 + 2·17-s − 8·19-s + 21-s + 8·23-s − 27-s − 6·29-s − 8·31-s − 33-s + 2·37-s − 6·39-s − 6·41-s − 4·43-s + 8·47-s + 49-s − 2·51-s + 10·53-s + 8·57-s + 2·61-s − 63-s − 4·67-s − 8·69-s + 4·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.485·17-s − 1.83·19-s + 0.218·21-s + 1.66·23-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.174·33-s + 0.328·37-s − 0.960·39-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 1.37·53-s + 1.05·57-s + 0.256·61-s − 0.125·63-s − 0.488·67-s − 0.963·69-s + 0.474·71-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

\(L(1)\) \(\approx\) \(1.931923657\)
\(L(\frac12)\) \(\approx\) \(1.931923657\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−12.64910423960486, −11.92496195473185, −11.65526070267681, −10.87350009704831, −10.83653563817214, −10.54328873526459, −9.765599930139171, −9.263359752226290, −8.851731177670734, −8.545100123691170, −7.970665446938471, −7.263495350773317, −6.908595940374645, −6.481182164822918, −6.015993200800382, −5.470824021754492, −5.252674936223965, −4.232926748873152, −4.126024984117241, −3.453556426395099, −3.071134138563723, −2.132521160087225, −1.700304763904454, −1.017108912336606, −0.4229766167465845, 0.4229766167465845, 1.017108912336606, 1.700304763904454, 2.132521160087225, 3.071134138563723, 3.453556426395099, 4.126024984117241, 4.232926748873152, 5.252674936223965, 5.470824021754492, 6.015993200800382, 6.481182164822918, 6.908595940374645, 7.263495350773317, 7.970665446938471, 8.545100123691170, 8.851731177670734, 9.263359752226290, 9.765599930139171, 10.54328873526459, 10.83653563817214, 10.87350009704831, 11.65526070267681, 11.92496195473185, 12.64910423960486

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