Properties

Label 2-3648-1.1-c1-0-13
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 + 2·5-s + 9-s − 6·13-s − 2·15-s − 6·17-s + 19-s + 4·23-s − 25-s − 27-s − 2·29-s + 8·31-s + 10·37-s + 6·39-s − 2·41-s + 4·43-s + 2·45-s + 12·47-s − 7·49-s + 6·51-s + 6·53-s − 57-s + 12·59-s + 2·61-s − 12·65-s + 4·67-s − 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 1.45·17-s + 0.229·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 1.64·37-s + 0.960·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.75·47-s − 49-s + 0.840·51-s + 0.824·53-s − 0.132·57-s + 1.56·59-s + 0.256·61-s − 1.48·65-s + 0.488·67-s − 0.481·69-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

\(L(1)\) \(\approx\) \(1.534849192\)
\(L(\frac12)\) \(\approx\) \(1.534849192\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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

−8.656147919084617168059664238609, −7.62692916507233066692617069633, −6.96313967889007715870809714116, −6.30766736214003560110551425788, −5.52157301147752958888078955745, −4.83501817913468223446739446530, −4.16209472915564587197687363565, −2.66586826806470965226318272179, −2.15027495389545596722403606493, −0.72964391205275968179204890320, 0.72964391205275968179204890320, 2.15027495389545596722403606493, 2.66586826806470965226318272179, 4.16209472915564587197687363565, 4.83501817913468223446739446530, 5.52157301147752958888078955745, 6.30766736214003560110551425788, 6.96313967889007715870809714116, 7.62692916507233066692617069633, 8.656147919084617168059664238609

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