Properties

Label 2-2205-1.1-c3-0-40
Degree $2$
Conductor $2205$
Sign $1$
Analytic cond. $130.099$
Root an. cond. $11.4061$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 4-s − 5·5-s + 21·8-s + 15·10-s + 24·11-s − 74·13-s − 71·16-s + 54·17-s + 124·19-s − 5·20-s − 72·22-s + 120·23-s + 25·25-s + 222·26-s + 78·29-s − 200·31-s + 45·32-s − 162·34-s − 70·37-s − 372·38-s − 105·40-s + 330·41-s + 92·43-s + 24·44-s − 360·46-s − 24·47-s + ⋯
L(s)  = 1  − 1.06·2-s + 1/8·4-s − 0.447·5-s + 0.928·8-s + 0.474·10-s + 0.657·11-s − 1.57·13-s − 1.10·16-s + 0.770·17-s + 1.49·19-s − 0.0559·20-s − 0.697·22-s + 1.08·23-s + 1/5·25-s + 1.67·26-s + 0.499·29-s − 1.15·31-s + 0.248·32-s − 0.817·34-s − 0.311·37-s − 1.58·38-s − 0.415·40-s + 1.25·41-s + 0.326·43-s + 0.0822·44-s − 1.15·46-s − 0.0744·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(130.099\)
Root analytic conductor: \(11.4061\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2205} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9379994835\)
\(L(\frac12)\) \(\approx\) \(0.9379994835\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
7 \( 1 \)
good2 \( 1 + 3 T + p^{3} T^{2} \)
11 \( 1 - 24 T + p^{3} T^{2} \)
13 \( 1 + 74 T + p^{3} T^{2} \)
17 \( 1 - 54 T + p^{3} T^{2} \)
19 \( 1 - 124 T + p^{3} T^{2} \)
23 \( 1 - 120 T + p^{3} T^{2} \)
29 \( 1 - 78 T + p^{3} T^{2} \)
31 \( 1 + 200 T + p^{3} T^{2} \)
37 \( 1 + 70 T + p^{3} T^{2} \)
41 \( 1 - 330 T + p^{3} T^{2} \)
43 \( 1 - 92 T + p^{3} T^{2} \)
47 \( 1 + 24 T + p^{3} T^{2} \)
53 \( 1 + 450 T + p^{3} T^{2} \)
59 \( 1 - 24 T + p^{3} T^{2} \)
61 \( 1 - 322 T + p^{3} T^{2} \)
67 \( 1 + 196 T + p^{3} T^{2} \)
71 \( 1 - 288 T + p^{3} T^{2} \)
73 \( 1 - 430 T + p^{3} T^{2} \)
79 \( 1 + 520 T + p^{3} T^{2} \)
83 \( 1 - 156 T + p^{3} T^{2} \)
89 \( 1 - 1026 T + p^{3} T^{2} \)
97 \( 1 - 286 T + p^{3} T^{2} \)
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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

−8.872183857462422900299313046457, −7.79316687641168669154280175365, −7.48282949202352947789234776692, −6.77096220357683672215251691479, −5.38744858113774301145412329362, −4.78717404912861731691706979711, −3.74705321285733907124172301911, −2.71506517231415179182608794675, −1.41146798293449679129928343468, −0.56600631635234203978422130342, 0.56600631635234203978422130342, 1.41146798293449679129928343468, 2.71506517231415179182608794675, 3.74705321285733907124172301911, 4.78717404912861731691706979711, 5.38744858113774301145412329362, 6.77096220357683672215251691479, 7.48282949202352947789234776692, 7.79316687641168669154280175365, 8.872183857462422900299313046457

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