Properties

Label 2-1638-1.1-c1-0-27
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 − 11-s − 13-s + 14-s + 16-s − 5·17-s − 19-s − 3·20-s − 22-s + 23-s + 4·25-s − 26-s + 28-s − 7·29-s − 2·31-s + 32-s − 5·34-s − 3·35-s + 37-s − 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 − 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.229·19-s − 0.670·20-s − 0.213·22-s + 0.208·23-s + 4/5·25-s − 0.196·26-s + 0.188·28-s − 1.29·29-s − 0.359·31-s + 0.176·32-s − 0.857·34-s − 0.507·35-s + 0.164·37-s − 0.162·38-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: 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.753348500037287827964908129272, −8.087399671123924969643801147843, −7.31743211521265601126957812633, −6.69061749873854041708189827769, −5.51931339869349074764387757967, −4.66162486191056383154167071598, −4.01680586816339694050615224274, −3.11346507231815676323880972931, −1.89913794199562214175475004949, 0, 1.89913794199562214175475004949, 3.11346507231815676323880972931, 4.01680586816339694050615224274, 4.66162486191056383154167071598, 5.51931339869349074764387757967, 6.69061749873854041708189827769, 7.31743211521265601126957812633, 8.087399671123924969643801147843, 8.753348500037287827964908129272

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