Properties

Label 2-1638-1.1-c1-0-22
Degree $2$
Conductor $1638$
Sign $-1$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 0.561·5-s − 7-s − 8-s − 0.561·10-s − 2.56·11-s + 13-s + 14-s + 16-s − 5.68·17-s + 7.68·19-s + 0.561·20-s + 2.56·22-s + 1.43·23-s − 4.68·25-s − 26-s − 28-s + 5.68·29-s − 10.2·31-s − 32-s + 5.68·34-s − 0.561·35-s − 3.43·37-s − 7.68·38-s − 0.561·40-s − 7.12·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.251·5-s − 0.377·7-s − 0.353·8-s − 0.177·10-s − 0.772·11-s + 0.277·13-s + 0.267·14-s + 0.250·16-s − 1.37·17-s + 1.76·19-s + 0.125·20-s + 0.546·22-s + 0.299·23-s − 0.936·25-s − 0.196·26-s − 0.188·28-s + 1.05·29-s − 1.84·31-s − 0.176·32-s + 0.974·34-s − 0.0949·35-s − 0.565·37-s − 1.24·38-s − 0.0887·40-s − 1.11·41-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: no
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 \( 1 - 0.561T + 5T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
17 \( 1 + 5.68T + 17T^{2} \)
19 \( 1 - 7.68T + 19T^{2} \)
23 \( 1 - 1.43T + 23T^{2} \)
29 \( 1 - 5.68T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 3.43T + 37T^{2} \)
41 \( 1 + 7.12T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 + 5.68T + 61T^{2} \)
67 \( 1 - 1.12T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 0.561T + 73T^{2} \)
79 \( 1 + 2.87T + 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−8.943546469941654705387114337009, −8.368753807190748414751207446163, −7.33181853461224455582846520339, −6.82486306202951059930453674231, −5.75786970631544080979429890362, −5.04332455074417045923424864103, −3.67375410635739582237562205909, −2.72240667229055867481077864761, −1.60297949560182135315821130298, 0, 1.60297949560182135315821130298, 2.72240667229055867481077864761, 3.67375410635739582237562205909, 5.04332455074417045923424864103, 5.75786970631544080979429890362, 6.82486306202951059930453674231, 7.33181853461224455582846520339, 8.368753807190748414751207446163, 8.943546469941654705387114337009

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