Properties

Label 2-2205-1.1-c1-0-45
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-s − 4-s + 5-s + 3·8-s − 10-s + 2·11-s − 6·13-s − 16-s − 6·17-s + 6·19-s − 20-s − 2·22-s + 4·23-s + 25-s + 6·26-s − 8·29-s + 6·31-s − 5·32-s + 6·34-s − 6·37-s − 6·38-s + 3·40-s − 6·41-s − 2·44-s − 4·46-s − 50-s + 6·52-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 0.603·11-s − 1.66·13-s − 1/4·16-s − 1.45·17-s + 1.37·19-s − 0.223·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s − 1.48·29-s + 1.07·31-s − 0.883·32-s + 1.02·34-s − 0.986·37-s − 0.973·38-s + 0.474·40-s − 0.937·41-s − 0.301·44-s − 0.589·46-s − 0.141·50-s + 0.832·52-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 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.929131819633424032481983386190, −7.952525594805363297976652459498, −7.21129932781375417432404118744, −6.58583184575665717452070168487, −5.21594225200877212023113542647, −4.86075960403071897586907134866, −3.75278266753238330801144288814, −2.50383922446396710571631521160, −1.42009345245993264838225109650, 0, 1.42009345245993264838225109650, 2.50383922446396710571631521160, 3.75278266753238330801144288814, 4.86075960403071897586907134866, 5.21594225200877212023113542647, 6.58583184575665717452070168487, 7.21129932781375417432404118744, 7.952525594805363297976652459498, 8.929131819633424032481983386190

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