Properties

Label 2-1638-1.1-c3-0-12
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 $0$

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 − 23·11-s − 13·13-s + 14·14-s + 16·16-s + 133·17-s + 103·19-s − 12·20-s + 46·22-s + 113·23-s − 116·25-s + 26·26-s − 28·28-s − 141·29-s − 90·31-s − 32·32-s − 266·34-s + 21·35-s − 233·37-s − 206·38-s + 24·40-s − 230·41-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 − 0.630·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.89·17-s + 1.24·19-s − 0.134·20-s + 0.445·22-s + 1.02·23-s − 0.927·25-s + 0.196·26-s − 0.188·28-s − 0.902·29-s − 0.521·31-s − 0.176·32-s − 1.34·34-s + 0.101·35-s − 1.03·37-s − 0.879·38-s + 0.0948·40-s − 0.876·41-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: $\chi_{1638} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.126728614\)
\(L(\frac12)\) \(\approx\) \(1.126728614\)
\(L(\frac{5}{2})\) 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 \( 1 + 23 T + p^{3} T^{2} \)
17 \( 1 - 133 T + p^{3} T^{2} \)
19 \( 1 - 103 T + p^{3} T^{2} \)
23 \( 1 - 113 T + p^{3} T^{2} \)
29 \( 1 + 141 T + p^{3} T^{2} \)
31 \( 1 + 90 T + p^{3} T^{2} \)
37 \( 1 + 233 T + p^{3} T^{2} \)
41 \( 1 + 230 T + p^{3} T^{2} \)
43 \( 1 + 337 T + p^{3} T^{2} \)
47 \( 1 - 96 T + p^{3} T^{2} \)
53 \( 1 + 134 T + p^{3} T^{2} \)
59 \( 1 - 838 T + p^{3} T^{2} \)
61 \( 1 - 295 T + p^{3} T^{2} \)
67 \( 1 - 468 T + p^{3} T^{2} \)
71 \( 1 - 54 T + p^{3} T^{2} \)
73 \( 1 + 553 T + p^{3} T^{2} \)
79 \( 1 + 694 T + p^{3} T^{2} \)
83 \( 1 + 1010 T + p^{3} T^{2} \)
89 \( 1 - 390 T + p^{3} T^{2} \)
97 \( 1 + 102 T + p^{3} T^{2} \)
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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

−9.052932653850584158373293697474, −8.170526922992171580972734470332, −7.46995904113873798005643058109, −6.96758004168410240256244335168, −5.63502053363970023579064403231, −5.22112646668366720339543693097, −3.61918191857082704504753588451, −3.04069392143738865524353185008, −1.70761156404040451926012849659, −0.57125336008409497087876502347, 0.57125336008409497087876502347, 1.70761156404040451926012849659, 3.04069392143738865524353185008, 3.61918191857082704504753588451, 5.22112646668366720339543693097, 5.63502053363970023579064403231, 6.96758004168410240256244335168, 7.46995904113873798005643058109, 8.170526922992171580972734470332, 9.052932653850584158373293697474

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