Properties

Label 2-2646-1.1-c1-0-42
Degree $2$
Conductor $2646$
Sign $-1$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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 + 8-s − 3·10-s − 3·11-s + 4·13-s + 16-s − 6·17-s + 7·19-s − 3·20-s − 3·22-s + 3·23-s + 4·25-s + 4·26-s − 5·31-s + 32-s − 6·34-s − 7·37-s + 7·38-s − 3·40-s − 9·41-s − 10·43-s − 3·44-s + 3·46-s + 6·47-s + 4·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s − 0.904·11-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 1.60·19-s − 0.670·20-s − 0.639·22-s + 0.625·23-s + 4/5·25-s + 0.784·26-s − 0.898·31-s + 0.176·32-s − 1.02·34-s − 1.15·37-s + 1.13·38-s − 0.474·40-s − 1.40·41-s − 1.52·43-s − 0.452·44-s + 0.442·46-s + 0.875·47-s + 0.565·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2646} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.411878978422576008118242977401, −7.54093826265732890551104966146, −7.07803356069366613643720682431, −6.12585114739427978407313474137, −5.13858233863725778435685317082, −4.55558448712373822916009028976, −3.51027276651897791446234749457, −3.11258133172613112703068504748, −1.63127189692849427124845257150, 0, 1.63127189692849427124845257150, 3.11258133172613112703068504748, 3.51027276651897791446234749457, 4.55558448712373822916009028976, 5.13858233863725778435685317082, 6.12585114739427978407313474137, 7.07803356069366613643720682431, 7.54093826265732890551104966146, 8.411878978422576008118242977401

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