Properties

Label 2-2646-1.1-c1-0-32
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 $0$

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 − 2·19-s + 3·20-s + 3·22-s + 6·23-s + 4·25-s + 4·26-s − 6·29-s − 5·31-s + 32-s + 2·37-s − 2·38-s + 3·40-s − 6·41-s − 10·43-s + 3·44-s + 6·46-s + 6·47-s + 4·50-s + 4·52-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 − 0.458·19-s + 0.670·20-s + 0.639·22-s + 1.25·23-s + 4/5·25-s + 0.784·26-s − 1.11·29-s − 0.898·31-s + 0.176·32-s + 0.328·37-s − 0.324·38-s + 0.474·40-s − 0.937·41-s − 1.52·43-s + 0.452·44-s + 0.884·46-s + 0.875·47-s + 0.565·50-s + 0.554·52-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.133596812\)
\(L(\frac12)\) \(\approx\) \(4.133596812\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−9.022301181159976542679631550139, −8.153752740732516759811873175243, −6.90390849159694968636551446179, −6.51972259556560194206176883369, −5.69537931970611061608929469001, −5.14293667469015758909535382312, −4.01126047180402972127592124525, −3.26671376711039705407751493003, −2.07276435338156744028852711093, −1.33346620954041525748246641205, 1.33346620954041525748246641205, 2.07276435338156744028852711093, 3.26671376711039705407751493003, 4.01126047180402972127592124525, 5.14293667469015758909535382312, 5.69537931970611061608929469001, 6.51972259556560194206176883369, 6.90390849159694968636551446179, 8.153752740732516759811873175243, 9.022301181159976542679631550139

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