Properties

Label 2-1638-1.1-c1-0-0
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 − 3·5-s − 7-s − 8-s + 3·10-s − 11-s − 13-s + 14-s + 16-s − 7·17-s + 19-s − 3·20-s + 22-s + 7·23-s + 4·25-s + 26-s − 28-s − 3·29-s − 32-s + 7·34-s + 3·35-s − 5·37-s − 38-s + 3·40-s − 4·41-s + 11·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s + 0.229·19-s − 0.670·20-s + 0.213·22-s + 1.45·23-s + 4/5·25-s + 0.196·26-s − 0.188·28-s − 0.557·29-s − 0.176·32-s + 1.20·34-s + 0.507·35-s − 0.821·37-s − 0.162·38-s + 0.474·40-s − 0.624·41-s + 1.67·43-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

\(L(1)\) \(\approx\) \(0.5875381626\)
\(L(\frac12)\) \(\approx\) \(0.5875381626\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−9.042633658738900996408374505805, −8.794485082046221789728545251895, −7.71886793500763207100130548275, −7.20481276136351597364288703843, −6.49587991264233814474632939114, −5.23206193743426850928425903169, −4.25945441857471440765324112942, −3.33711352613587716311643994647, −2.28508171651994554248500999764, −0.56722170613835914134961311701, 0.56722170613835914134961311701, 2.28508171651994554248500999764, 3.33711352613587716311643994647, 4.25945441857471440765324112942, 5.23206193743426850928425903169, 6.49587991264233814474632939114, 7.20481276136351597364288703843, 7.71886793500763207100130548275, 8.794485082046221789728545251895, 9.042633658738900996408374505805

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