Properties

Label 2-2205-1.1-c1-0-41
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5-s + 3·11-s − 5·13-s + 4·16-s + 3·17-s − 2·19-s + 2·20-s + 6·23-s + 25-s − 3·29-s + 4·31-s + 2·37-s − 12·41-s − 10·43-s − 6·44-s + 9·47-s + 10·52-s − 12·53-s − 3·55-s − 8·61-s − 8·64-s + 5·65-s − 4·67-s − 6·68-s − 2·73-s + 4·76-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s + 0.904·11-s − 1.38·13-s + 16-s + 0.727·17-s − 0.458·19-s + 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.557·29-s + 0.718·31-s + 0.328·37-s − 1.87·41-s − 1.52·43-s − 0.904·44-s + 1.31·47-s + 1.38·52-s − 1.64·53-s − 0.404·55-s − 1.02·61-s − 64-s + 0.620·65-s − 0.488·67-s − 0.727·68-s − 0.234·73-s + 0.458·76-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: \(1\)
Selberg data: \((2,\ 2205,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−8.727723074824453204124206103229, −7.959921722640743022302973334932, −7.21456919347212808652226059968, −6.36641447795918613892295316962, −5.19374718417958536543626274426, −4.71225653823478357730435111839, −3.78858238344580500375502888439, −2.94585098531436589990293317610, −1.37785913473779166712390600334, 0, 1.37785913473779166712390600334, 2.94585098531436589990293317610, 3.78858238344580500375502888439, 4.71225653823478357730435111839, 5.19374718417958536543626274426, 6.36641447795918613892295316962, 7.21456919347212808652226059968, 7.959921722640743022302973334932, 8.727723074824453204124206103229

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