Properties

Label 2-1638-1.1-c1-0-15
Degree $2$
Conductor $1638$
Sign $1$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 4·11-s + 13-s − 14-s + 16-s + 2·17-s − 4·19-s + 2·20-s + 4·22-s + 4·23-s − 25-s + 26-s − 28-s + 2·29-s + 32-s + 2·34-s − 2·35-s − 2·37-s − 4·38-s + 2·40-s − 2·41-s + 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 1.20·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.188·28-s + 0.371·29-s + 0.176·32-s + 0.342·34-s − 0.338·35-s − 0.328·37-s − 0.648·38-s + 0.316·40-s − 0.312·41-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.315613366\)
\(L(\frac12)\) \(\approx\) \(3.315613366\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
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 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
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

−9.344633920829121985006795507809, −8.793062426761640380192286442759, −7.60950019354364822141375551305, −6.62410620605336828325420319307, −6.18971912035243321530580658857, −5.36734378010452113216798273238, −4.32571873646081315893153040604, −3.48844535790283312124632987251, −2.39519789676916156425539342752, −1.29507569141715195434836244797, 1.29507569141715195434836244797, 2.39519789676916156425539342752, 3.48844535790283312124632987251, 4.32571873646081315893153040604, 5.36734378010452113216798273238, 6.18971912035243321530580658857, 6.62410620605336828325420319307, 7.60950019354364822141375551305, 8.793062426761640380192286442759, 9.344633920829121985006795507809

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