Properties

Label 2-3648-1.1-c1-0-36
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 + 4·13-s + 6·17-s + 19-s + 4·21-s + 6·23-s − 5·25-s + 27-s − 6·29-s − 2·31-s + 4·37-s + 4·39-s + 6·41-s − 4·43-s − 6·47-s + 9·49-s + 6·51-s − 6·53-s + 57-s − 12·59-s − 14·61-s + 4·63-s + 8·67-s + 6·69-s + 14·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s + 0.229·19-s + 0.872·21-s + 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.657·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.875·47-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 0.132·57-s − 1.56·59-s − 1.79·61-s + 0.503·63-s + 0.977·67-s + 0.722·69-s + 1.63·73-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: $\chi_{3648} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.273141381\)
\(L(\frac12)\) \(\approx\) \(3.273141381\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−8.380140351181000236453607755385, −7.83146723424270026458224882246, −7.43283795707002150488374868194, −6.23636375738531983020866481834, −5.45716303902372547637431694432, −4.74514961136841231763588047323, −3.81199005087099183295491504054, −3.09156700676949380226753412779, −1.82297840574153350889823289813, −1.18152128549559051518186862695, 1.18152128549559051518186862695, 1.82297840574153350889823289813, 3.09156700676949380226753412779, 3.81199005087099183295491504054, 4.74514961136841231763588047323, 5.45716303902372547637431694432, 6.23636375738531983020866481834, 7.43283795707002150488374868194, 7.83146723424270026458224882246, 8.380140351181000236453607755385

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