Properties

Label 2-3648-1.1-c1-0-41
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 − 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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
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

−7.992680376128817439410616241148, −7.47620423165504362047840493205, −6.37837332665094581753674197922, −6.03723114482210006778449723144, −5.45803980005957483798159176539, −4.03060027612472363140407850727, −3.67844472597115610426331813053, −2.59451510562992864284961947324, −1.24718934600363714757600412295, 0, 1.24718934600363714757600412295, 2.59451510562992864284961947324, 3.67844472597115610426331813053, 4.03060027612472363140407850727, 5.45803980005957483798159176539, 6.03723114482210006778449723144, 6.37837332665094581753674197922, 7.47620423165504362047840493205, 7.992680376128817439410616241148

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