Properties

Label 2-1638-1.1-c1-0-28
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 − 5-s − 7-s + 8-s − 10-s − 5·11-s + 13-s − 14-s + 16-s − 17-s − 19-s − 20-s − 5·22-s − 5·23-s − 4·25-s + 26-s − 28-s − 29-s − 6·31-s + 32-s − 34-s + 35-s + 7·37-s − 38-s − 40-s − 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 1.50·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.229·19-s − 0.223·20-s − 1.06·22-s − 1.04·23-s − 4/5·25-s + 0.196·26-s − 0.188·28-s − 0.185·29-s − 1.07·31-s + 0.176·32-s − 0.171·34-s + 0.169·35-s + 1.15·37-s − 0.162·38-s − 0.158·40-s − 0.312·41-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 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.951187799408442413225861722767, −7.83195642869312453580393936381, −7.58771647878142615673084205238, −6.34336856013115106006521996723, −5.72816854709225568428895208685, −4.77808523077732043316959668397, −3.91582934000479521066630000171, −3.00954150321325090761034987812, −1.99707427963496596135030374522, 0, 1.99707427963496596135030374522, 3.00954150321325090761034987812, 3.91582934000479521066630000171, 4.77808523077732043316959668397, 5.72816854709225568428895208685, 6.34336856013115106006521996723, 7.58771647878142615673084205238, 7.83195642869312453580393936381, 8.951187799408442413225861722767

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