Properties

Label 2-1638-1.1-c1-0-26
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 $1$

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 − 3·11-s + 13-s + 14-s + 16-s + 3·17-s − 7·19-s − 3·20-s − 3·22-s − 9·23-s + 4·25-s + 26-s + 28-s + 9·29-s − 4·31-s + 32-s + 3·34-s − 3·35-s − 7·37-s − 7·38-s − 3·40-s − 12·41-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 − 0.904·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.60·19-s − 0.670·20-s − 0.639·22-s − 1.87·23-s + 4/5·25-s + 0.196·26-s + 0.188·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s − 0.507·35-s − 1.15·37-s − 1.13·38-s − 0.474·40-s − 1.87·41-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: \(1\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
7 \( 1 - T \)
13 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−8.494657990807389898658506219562, −8.231691217522472057103385997568, −7.42970330921915709586726435368, −6.54796581741576703271548356667, −5.59399109559353101448430855804, −4.65012025043354629223454819364, −3.98883124750352014007177515851, −3.13495261897234347757421228131, −1.89338424078267539825922227519, 0, 1.89338424078267539825922227519, 3.13495261897234347757421228131, 3.98883124750352014007177515851, 4.65012025043354629223454819364, 5.59399109559353101448430855804, 6.54796581741576703271548356667, 7.42970330921915709586726435368, 8.231691217522472057103385997568, 8.494657990807389898658506219562

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