Properties

Label 2-58800-1.1-c1-0-34
Degree $2$
Conductor $58800$
Sign $1$
Analytic cond. $469.520$
Root an. cond. $21.6684$
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 + 9-s − 4·11-s − 2·13-s − 2·17-s − 2·19-s + 6·23-s − 27-s + 6·29-s + 6·31-s + 4·33-s − 4·37-s + 2·39-s + 4·43-s + 4·47-s + 2·51-s − 2·53-s + 2·57-s + 4·59-s + 2·61-s + 12·67-s − 6·69-s + 8·71-s − 14·73-s − 16·79-s + 81-s + 16·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.458·19-s + 1.25·23-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.696·33-s − 0.657·37-s + 0.320·39-s + 0.609·43-s + 0.583·47-s + 0.280·51-s − 0.274·53-s + 0.264·57-s + 0.520·59-s + 0.256·61-s + 1.46·67-s − 0.722·69-s + 0.949·71-s − 1.63·73-s − 1.80·79-s + 1/9·81-s + 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(469.520\)
Root analytic conductor: \(21.6684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{58800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.266289480\)
\(L(\frac12)\) \(\approx\) \(1.266289480\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−14.34106078864868, −13.73382749741617, −13.26530354780711, −12.74267525579589, −12.40913653179873, −11.78866237269192, −11.23677470765260, −10.76422562187317, −10.27409889658543, −9.933230906281547, −9.174265864254173, −8.613002021187189, −8.101712113925828, −7.519340750297129, −6.831346046841964, −6.613944462021226, −5.698178011948714, −5.328339920731170, −4.651629790057131, −4.361836876209688, −3.352603046079798, −2.671593920176278, −2.239637833828222, −1.179326012554634, −0.4328616827527743, 0.4328616827527743, 1.179326012554634, 2.239637833828222, 2.671593920176278, 3.352603046079798, 4.361836876209688, 4.651629790057131, 5.328339920731170, 5.698178011948714, 6.613944462021226, 6.831346046841964, 7.519340750297129, 8.101712113925828, 8.613002021187189, 9.174265864254173, 9.933230906281547, 10.27409889658543, 10.76422562187317, 11.23677470765260, 11.78866237269192, 12.40913653179873, 12.74267525579589, 13.26530354780711, 13.73382749741617, 14.34106078864868

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