Properties

Label 2-1638-1.1-c3-0-64
Degree $2$
Conductor $1638$
Sign $-1$
Analytic cond. $96.6451$
Root an. cond. $9.83082$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s − 3·5-s + 7·7-s − 8·8-s + 6·10-s + 54·11-s + 13·13-s − 14·14-s + 16·16-s + 96·17-s − 151·19-s − 12·20-s − 108·22-s − 33·23-s − 116·25-s − 26·26-s + 28·28-s − 183·29-s − 331·31-s − 32·32-s − 192·34-s − 21·35-s − 88·37-s + 302·38-s + 24·40-s + 42·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.268·5-s + 0.377·7-s − 0.353·8-s + 0.189·10-s + 1.48·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.36·17-s − 1.82·19-s − 0.134·20-s − 1.04·22-s − 0.299·23-s − 0.927·25-s − 0.196·26-s + 0.188·28-s − 1.17·29-s − 1.91·31-s − 0.176·32-s − 0.968·34-s − 0.101·35-s − 0.391·37-s + 1.28·38-s + 0.0948·40-s + 0.159·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(96.6451\)
Root analytic conductor: \(9.83082\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1638,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 \)
7 \( 1 - p T \)
13 \( 1 - p T \)
good5 \( 1 + 3 T + p^{3} T^{2} \)
11 \( 1 - 54 T + p^{3} T^{2} \)
17 \( 1 - 96 T + p^{3} T^{2} \)
19 \( 1 + 151 T + p^{3} T^{2} \)
23 \( 1 + 33 T + p^{3} T^{2} \)
29 \( 1 + 183 T + p^{3} T^{2} \)
31 \( 1 + 331 T + p^{3} T^{2} \)
37 \( 1 + 88 T + p^{3} T^{2} \)
41 \( 1 - 42 T + p^{3} T^{2} \)
43 \( 1 - 353 T + p^{3} T^{2} \)
47 \( 1 - 465 T + p^{3} T^{2} \)
53 \( 1 + 195 T + p^{3} T^{2} \)
59 \( 1 + 552 T + p^{3} T^{2} \)
61 \( 1 - 470 T + p^{3} T^{2} \)
67 \( 1 - 254 T + p^{3} T^{2} \)
71 \( 1 + 132 T + p^{3} T^{2} \)
73 \( 1 + 943 T + p^{3} T^{2} \)
79 \( 1 + 727 T + p^{3} T^{2} \)
83 \( 1 - 1197 T + p^{3} T^{2} \)
89 \( 1 + 753 T + p^{3} T^{2} \)
97 \( 1 - 1037 T + p^{3} T^{2} \)
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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

−8.807778038715973021333280679946, −7.79719022622809512024979568894, −7.26926588472458538073084200229, −6.22045101214396853082940826840, −5.61466368032343484444658193040, −4.14001728350096255951137313476, −3.63998825054373724883435038736, −2.10844303475684512184237462396, −1.30600100397476910703189075135, 0, 1.30600100397476910703189075135, 2.10844303475684512184237462396, 3.63998825054373724883435038736, 4.14001728350096255951137313476, 5.61466368032343484444658193040, 6.22045101214396853082940826840, 7.26926588472458538073084200229, 7.79719022622809512024979568894, 8.807778038715973021333280679946

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