Properties

Label 2-1638-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 4·5-s − 7-s − 8-s − 4·10-s − 11-s − 13-s + 14-s + 16-s − 6·19-s + 4·20-s + 22-s + 7·23-s + 11·25-s + 26-s − 28-s + 4·29-s + 7·31-s − 32-s − 4·35-s + 9·37-s + 6·38-s − 4·40-s + 3·41-s + 4·43-s − 44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.377·7-s − 0.353·8-s − 1.26·10-s − 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.37·19-s + 0.894·20-s + 0.213·22-s + 1.45·23-s + 11/5·25-s + 0.196·26-s − 0.188·28-s + 0.742·29-s + 1.25·31-s − 0.176·32-s − 0.676·35-s + 1.47·37-s + 0.973·38-s − 0.632·40-s + 0.468·41-s + 0.609·43-s − 0.150·44-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: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.650233464\)
\(L(\frac12)\) \(\approx\) \(1.650233464\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−9.382683634490519654764199003796, −8.842134311662064724454282147488, −7.943345217051475117643242740083, −6.72942118020662831970006205526, −6.37840271109207786796552495603, −5.49386334296658824475642157597, −4.55256625494296783763579353487, −2.85733179130965228288861874054, −2.28863422577859859560688974167, −1.03297262712619069700335338528, 1.03297262712619069700335338528, 2.28863422577859859560688974167, 2.85733179130965228288861874054, 4.55256625494296783763579353487, 5.49386334296658824475642157597, 6.37840271109207786796552495603, 6.72942118020662831970006205526, 7.943345217051475117643242740083, 8.842134311662064724454282147488, 9.382683634490519654764199003796

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