Properties

Label 2-3648-1.1-c1-0-69
Degree $2$
Conductor $3648$
Sign $-1$
Analytic cond. $29.1294$
Root an. cond. $5.39716$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 2·17-s + 19-s − 6·23-s − 5·25-s + 27-s − 10·29-s − 10·31-s − 8·37-s + 6·41-s − 4·43-s + 6·47-s − 7·49-s − 2·51-s + 6·53-s + 57-s + 12·59-s + 10·61-s + 8·67-s − 6·69-s − 8·71-s − 2·73-s − 5·75-s − 6·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.485·17-s + 0.229·19-s − 1.25·23-s − 25-s + 0.192·27-s − 1.85·29-s − 1.79·31-s − 1.31·37-s + 0.937·41-s − 0.609·43-s + 0.875·47-s − 49-s − 0.280·51-s + 0.824·53-s + 0.132·57-s + 1.56·59-s + 1.28·61-s + 0.977·67-s − 0.722·69-s − 0.949·71-s − 0.234·73-s − 0.577·75-s − 0.675·79-s + 1/9·81-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $-1$
Analytic conductor: \(29.1294\)
Root analytic conductor: \(5.39716\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3648} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3648,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−8.168970345890267096694431770502, −7.46673926194569887500692717232, −6.88565694981038390632634000087, −5.80792412759310459185090671492, −5.27011383596716056041376055900, −3.93470913415548969633528171479, −3.72465779285701629479354216953, −2.36515606401106323529575170328, −1.71729500698904191815053448386, 0, 1.71729500698904191815053448386, 2.36515606401106323529575170328, 3.72465779285701629479354216953, 3.93470913415548969633528171479, 5.27011383596716056041376055900, 5.80792412759310459185090671492, 6.88565694981038390632634000087, 7.46673926194569887500692717232, 8.168970345890267096694431770502

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