Properties

Label 2-1638-1.1-c1-0-18
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 + 5·19-s − 3·20-s + 3·22-s + 3·23-s + 4·25-s − 26-s + 28-s + 3·29-s − 10·31-s − 32-s − 3·34-s − 3·35-s − 7·37-s − 5·38-s + 3·40-s − 6·41-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 + 1.14·19-s − 0.670·20-s + 0.639·22-s + 0.625·23-s + 4/5·25-s − 0.196·26-s + 0.188·28-s + 0.557·29-s − 1.79·31-s − 0.176·32-s − 0.514·34-s − 0.507·35-s − 1.15·37-s − 0.811·38-s + 0.474·40-s − 0.937·41-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: $\chi_{1638} (1, \cdot )$
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 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.851633452241949887706170327808, −8.091285142177144740601668535575, −7.56353523982414710439528855608, −7.01142907917938592333952322993, −5.65940843144591511168796809859, −4.89898920843487998926105208871, −3.68091448723354985781129327882, −2.95150623961030889311106388300, −1.40498596438442826877223932759, 0, 1.40498596438442826877223932759, 2.95150623961030889311106388300, 3.68091448723354985781129327882, 4.89898920843487998926105208871, 5.65940843144591511168796809859, 7.01142907917938592333952322993, 7.56353523982414710439528855608, 8.091285142177144740601668535575, 8.851633452241949887706170327808

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