Properties

Label 2-2205-1.1-c1-0-20
Degree $2$
Conductor $2205$
Sign $1$
Analytic cond. $17.6070$
Root an. cond. $4.19607$
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 + 5-s + 3·8-s − 10-s + 6·13-s − 16-s + 2·17-s + 8·19-s − 20-s − 8·23-s + 25-s − 6·26-s + 2·29-s − 4·31-s − 5·32-s − 2·34-s − 2·37-s − 8·38-s + 3·40-s − 6·41-s + 4·43-s + 8·46-s + 8·47-s − 50-s − 6·52-s − 10·53-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 1.66·13-s − 1/4·16-s + 0.485·17-s + 1.83·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s − 1.17·26-s + 0.371·29-s − 0.718·31-s − 0.883·32-s − 0.342·34-s − 0.328·37-s − 1.29·38-s + 0.474·40-s − 0.937·41-s + 0.609·43-s + 1.17·46-s + 1.16·47-s − 0.141·50-s − 0.832·52-s − 1.37·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(17.6070\)
Root analytic conductor: \(4.19607\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2205} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.279510442\)
\(L(\frac12)\) \(\approx\) \(1.279510442\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−9.158915116725801964521352538031, −8.275102727179600443330310060644, −7.82733728798443659692174656726, −6.82600380115164878858049675491, −5.81605182603986018580602603999, −5.25448844749250196408410433194, −4.05504421439464721652879145719, −3.34833988846054044432199186298, −1.79849192746842450415563444508, −0.888154170390839902230274080915, 0.888154170390839902230274080915, 1.79849192746842450415563444508, 3.34833988846054044432199186298, 4.05504421439464721652879145719, 5.25448844749250196408410433194, 5.81605182603986018580602603999, 6.82600380115164878858049675491, 7.82733728798443659692174656726, 8.275102727179600443330310060644, 9.158915116725801964521352538031

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