Properties

Label 2-2205-1.1-c1-0-17
Degree $2$
Conductor $2205$
Sign $1$
Analytic cond. $17.6070$
Root an. cond. $4.19607$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s − 5-s + 2.82·8-s + 1.41·10-s + 5.82·11-s + 1.58·13-s − 4.00·16-s + 5.24·17-s + 6·19-s − 8.24·22-s − 4.58·23-s + 25-s − 2.24·26-s − 2.65·29-s + 1.75·31-s − 7.41·34-s − 6.24·37-s − 8.48·38-s − 2.82·40-s − 2.24·41-s + 2·43-s + 6.48·46-s − 1.24·47-s − 1.41·50-s + ⋯
L(s)  = 1  − 1.00·2-s − 0.447·5-s + 0.999·8-s + 0.447·10-s + 1.75·11-s + 0.439·13-s − 1.00·16-s + 1.27·17-s + 1.37·19-s − 1.75·22-s − 0.956·23-s + 0.200·25-s − 0.439·26-s − 0.493·29-s + 0.315·31-s − 1.27·34-s − 1.02·37-s − 1.37·38-s − 0.447·40-s − 0.350·41-s + 0.304·43-s + 0.956·46-s − 0.181·47-s − 0.200·50-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: no
Arithmetic: yes
Character: $\chi_{2205} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.047070702\)
\(L(\frac12)\) \(\approx\) \(1.047070702\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41T + 2T^{2} \)
11 \( 1 - 5.82T + 11T^{2} \)
13 \( 1 - 1.58T + 13T^{2} \)
17 \( 1 - 5.24T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 4.58T + 23T^{2} \)
29 \( 1 + 2.65T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 + 6.24T + 37T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 1.24T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 6.24T + 59T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 - 0.242T + 67T^{2} \)
71 \( 1 - 8.82T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 - 4.75T + 97T^{2} \)
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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

−9.071909591690618624898709556636, −8.379991780454674846004898979313, −7.65077403859023603295446257660, −7.02394662091311261757133788545, −6.04767864297534456620417626782, −5.06215173758793164073558378422, −3.97851752142182424656321860884, −3.42771083589053003477445781080, −1.66663754915808709059767631528, −0.856417933321485659184945988999, 0.856417933321485659184945988999, 1.66663754915808709059767631528, 3.42771083589053003477445781080, 3.97851752142182424656321860884, 5.06215173758793164073558378422, 6.04767864297534456620417626782, 7.02394662091311261757133788545, 7.65077403859023603295446257660, 8.379991780454674846004898979313, 9.071909591690618624898709556636

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