Properties

Label 2-369600-1.1-c1-0-123
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 − 2·17-s + 4·19-s − 21-s − 6·23-s − 27-s + 4·29-s + 10·31-s + 33-s + 8·37-s + 6·41-s − 12·43-s − 8·47-s + 49-s + 2·51-s + 6·53-s − 4·57-s − 8·59-s − 2·61-s + 63-s + 8·67-s + 6·69-s − 8·71-s + 8·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s + 0.742·29-s + 1.79·31-s + 0.174·33-s + 1.31·37-s + 0.937·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s − 1.04·59-s − 0.256·61-s + 0.125·63-s + 0.977·67-s + 0.722·69-s − 0.949·71-s + 0.936·73-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

\(L(1)\) \(\approx\) \(2.020502888\)
\(L(\frac12)\) \(\approx\) \(2.020502888\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−12.44771177820175, −11.92105742366390, −11.62441084946268, −11.24811805383227, −10.77173899585496, −10.11910996661720, −9.897102743469534, −9.550317395808048, −8.774199173980571, −8.266268960792188, −8.006309270690529, −7.479348640984145, −6.923080208926628, −6.333949526600284, −6.122982245861292, −5.499871830387154, −4.918294215358643, −4.590478151009753, −4.141133233695286, −3.416721490520795, −2.855841012580426, −2.304879475689160, −1.656326796528573, −1.015678672410970, −0.4324198938906397, 0.4324198938906397, 1.015678672410970, 1.656326796528573, 2.304879475689160, 2.855841012580426, 3.416721490520795, 4.141133233695286, 4.590478151009753, 4.918294215358643, 5.499871830387154, 6.122982245861292, 6.333949526600284, 6.923080208926628, 7.479348640984145, 8.006309270690529, 8.266268960792188, 8.774199173980571, 9.550317395808048, 9.897102743469534, 10.11910996661720, 10.77173899585496, 11.24811805383227, 11.62441084946268, 11.92105742366390, 12.44771177820175

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