Properties

Label 2-369600-1.1-c1-0-13
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 + 4·23-s − 27-s + 2·29-s − 4·31-s − 33-s − 2·37-s + 2·39-s − 2·41-s − 12·43-s + 49-s − 6·51-s − 2·53-s + 4·57-s + 14·61-s − 63-s − 8·67-s − 4·69-s − 16·71-s − 10·73-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 + 0.834·23-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 − 0.312·41-s − 1.82·43-s + 1/7·49-s − 0.840·51-s − 0.274·53-s + 0.529·57-s + 1.79·61-s − 0.125·63-s − 0.977·67-s − 0.481·69-s − 1.89·71-s − 1.17·73-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

\(L(1)\) \(\approx\) \(0.5349290555\)
\(L(\frac12)\) \(\approx\) \(0.5349290555\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.38528729542293, −12.12053002410792, −11.60311148498665, −11.25636950195244, −10.60324609053894, −10.21079299228438, −9.943143451576450, −9.424264679608376, −8.825877441376743, −8.474403670374517, −7.888788623496416, −7.234898054692980, −7.081353917179076, −6.415010430124998, −6.057415646303130, −5.474077320609462, −5.000572485222939, −4.677324833386943, −3.833252367706312, −3.565910596332073, −2.889419533073189, −2.362343128364787, −1.493324543942802, −1.193994544645858, −0.2035181319225544, 0.2035181319225544, 1.193994544645858, 1.493324543942802, 2.362343128364787, 2.889419533073189, 3.565910596332073, 3.833252367706312, 4.677324833386943, 5.000572485222939, 5.474077320609462, 6.057415646303130, 6.415010430124998, 7.081353917179076, 7.234898054692980, 7.888788623496416, 8.474403670374517, 8.825877441376743, 9.424264679608376, 9.943143451576450, 10.21079299228438, 10.60324609053894, 11.25636950195244, 11.60311148498665, 12.12053002410792, 12.38528729542293

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