Properties

Label 2-3648-1.1-c1-0-21
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4·7-s + 9-s − 4·11-s − 2·17-s − 19-s + 4·21-s − 2·23-s − 5·25-s + 27-s + 6·29-s + 6·31-s − 4·33-s + 8·37-s + 10·41-s + 12·43-s + 10·47-s + 9·49-s − 2·51-s − 2·53-s − 57-s − 4·59-s + 10·61-s + 4·63-s − 2·69-s − 16·71-s − 2·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.485·17-s − 0.229·19-s + 0.872·21-s − 0.417·23-s − 25-s + 0.192·27-s + 1.11·29-s + 1.07·31-s − 0.696·33-s + 1.31·37-s + 1.56·41-s + 1.82·43-s + 1.45·47-s + 9/7·49-s − 0.280·51-s − 0.274·53-s − 0.132·57-s − 0.520·59-s + 1.28·61-s + 0.503·63-s − 0.240·69-s − 1.89·71-s − 0.234·73-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.699634895\)
\(L(\frac12)\) \(\approx\) \(2.699634895\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−8.358505217531610024473691264975, −7.83554718782563819072679671534, −7.46691861053528939766555450107, −6.22898639923753537168460591215, −5.48900411534536167558093009531, −4.52705265518562875166169538319, −4.17017825141943436906107680698, −2.66003526278717564626965589299, −2.24707632442142343769759243716, −0.963363230503270302191434620474, 0.963363230503270302191434620474, 2.24707632442142343769759243716, 2.66003526278717564626965589299, 4.17017825141943436906107680698, 4.52705265518562875166169538319, 5.48900411534536167558093009531, 6.22898639923753537168460591215, 7.46691861053528939766555450107, 7.83554718782563819072679671534, 8.358505217531610024473691264975

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