Properties

Label 2-2205-1.1-c3-0-112
Degree $2$
Conductor $2205$
Sign $-1$
Analytic cond. $130.099$
Root an. cond. $11.4061$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.09·2-s + 1.60·4-s − 5·5-s + 19.8·8-s + 15.4·10-s + 31.1·11-s − 54.1·13-s − 74.2·16-s + 81.8·17-s + 37.3·19-s − 8.03·20-s − 96.6·22-s − 116.·23-s + 25·25-s + 167.·26-s − 22.5·29-s − 89.9·31-s + 71.6·32-s − 253.·34-s + 344.·37-s − 115.·38-s − 99.0·40-s − 245.·41-s + 41.4·43-s + 50.1·44-s + 360.·46-s − 431.·47-s + ⋯
L(s)  = 1  − 1.09·2-s + 0.200·4-s − 0.447·5-s + 0.875·8-s + 0.490·10-s + 0.854·11-s − 1.15·13-s − 1.16·16-s + 1.16·17-s + 0.451·19-s − 0.0898·20-s − 0.936·22-s − 1.05·23-s + 0.200·25-s + 1.26·26-s − 0.144·29-s − 0.521·31-s + 0.395·32-s − 1.28·34-s + 1.53·37-s − 0.494·38-s − 0.391·40-s − 0.936·41-s + 0.147·43-s + 0.171·44-s + 1.15·46-s − 1.33·47-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: no
Arithmetic: yes
Character: $\chi_{2205} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2205,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 5T \)
7 \( 1 \)
good2 \( 1 + 3.09T + 8T^{2} \)
11 \( 1 - 31.1T + 1.33e3T^{2} \)
13 \( 1 + 54.1T + 2.19e3T^{2} \)
17 \( 1 - 81.8T + 4.91e3T^{2} \)
19 \( 1 - 37.3T + 6.85e3T^{2} \)
23 \( 1 + 116.T + 1.21e4T^{2} \)
29 \( 1 + 22.5T + 2.43e4T^{2} \)
31 \( 1 + 89.9T + 2.97e4T^{2} \)
37 \( 1 - 344.T + 5.06e4T^{2} \)
41 \( 1 + 245.T + 6.89e4T^{2} \)
43 \( 1 - 41.4T + 7.95e4T^{2} \)
47 \( 1 + 431.T + 1.03e5T^{2} \)
53 \( 1 - 263.T + 1.48e5T^{2} \)
59 \( 1 + 627.T + 2.05e5T^{2} \)
61 \( 1 - 715.T + 2.26e5T^{2} \)
67 \( 1 + 809.T + 3.00e5T^{2} \)
71 \( 1 + 54.8T + 3.57e5T^{2} \)
73 \( 1 - 508.T + 3.89e5T^{2} \)
79 \( 1 - 61.8T + 4.93e5T^{2} \)
83 \( 1 + 560.T + 5.71e5T^{2} \)
89 \( 1 + 102.T + 7.04e5T^{2} \)
97 \( 1 + 253.T + 9.12e5T^{2} \)
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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

−8.303665084589683232427006584620, −7.65382217386887990980323040037, −7.17251252683292583876568568260, −6.10877461884141117853217637960, −5.04989761406060344504865915257, −4.24565769716126065642021892730, −3.30294695709990322095155745643, −1.99556019850539292409153997101, −1.00033382417056716184867423296, 0, 1.00033382417056716184867423296, 1.99556019850539292409153997101, 3.30294695709990322095155745643, 4.24565769716126065642021892730, 5.04989761406060344504865915257, 6.10877461884141117853217637960, 7.17251252683292583876568568260, 7.65382217386887990980323040037, 8.303665084589683232427006584620

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