Properties

Label 2-2205-1.1-c3-0-123
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  − 4.46·2-s + 11.9·4-s − 5·5-s − 17.6·8-s + 22.3·10-s + 56.5·11-s − 40.9·13-s − 16.6·16-s + 2.18·17-s − 16.4·19-s − 59.7·20-s − 252.·22-s + 155.·23-s + 25·25-s + 183.·26-s + 6.26·29-s − 168.·31-s + 215.·32-s − 9.77·34-s − 37.1·37-s + 73.5·38-s + 88.4·40-s − 266.·41-s − 14.6·43-s + 676.·44-s − 693.·46-s − 169.·47-s + ⋯
L(s)  = 1  − 1.57·2-s + 1.49·4-s − 0.447·5-s − 0.781·8-s + 0.706·10-s + 1.54·11-s − 0.873·13-s − 0.260·16-s + 0.0312·17-s − 0.198·19-s − 0.668·20-s − 2.44·22-s + 1.40·23-s + 0.200·25-s + 1.38·26-s + 0.0400·29-s − 0.977·31-s + 1.19·32-s − 0.0493·34-s − 0.165·37-s + 0.314·38-s + 0.349·40-s − 1.01·41-s − 0.0519·43-s + 2.31·44-s − 2.22·46-s − 0.527·47-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: Trivial
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 \( 1 + 4.46T + 8T^{2} \)
11 \( 1 - 56.5T + 1.33e3T^{2} \)
13 \( 1 + 40.9T + 2.19e3T^{2} \)
17 \( 1 - 2.18T + 4.91e3T^{2} \)
19 \( 1 + 16.4T + 6.85e3T^{2} \)
23 \( 1 - 155.T + 1.21e4T^{2} \)
29 \( 1 - 6.26T + 2.43e4T^{2} \)
31 \( 1 + 168.T + 2.97e4T^{2} \)
37 \( 1 + 37.1T + 5.06e4T^{2} \)
41 \( 1 + 266.T + 6.89e4T^{2} \)
43 \( 1 + 14.6T + 7.95e4T^{2} \)
47 \( 1 + 169.T + 1.03e5T^{2} \)
53 \( 1 - 151.T + 1.48e5T^{2} \)
59 \( 1 + 234.T + 2.05e5T^{2} \)
61 \( 1 - 242.T + 2.26e5T^{2} \)
67 \( 1 - 820.T + 3.00e5T^{2} \)
71 \( 1 + 961.T + 3.57e5T^{2} \)
73 \( 1 + 934.T + 3.89e5T^{2} \)
79 \( 1 - 300.T + 4.93e5T^{2} \)
83 \( 1 - 1.08e3T + 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 752.T + 9.12e5T^{2} \)
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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

−8.577648669701098265484375790233, −7.58409654765888915337872268977, −7.03094791753694980624143862332, −6.45472420143965209062030297420, −5.13839343490042737674841580871, −4.17027690025449907385227268342, −3.10807392973969823565264596720, −1.89957176120048948966404910956, −1.03782097414346143926121900093, 0, 1.03782097414346143926121900093, 1.89957176120048948966404910956, 3.10807392973969823565264596720, 4.17027690025449907385227268342, 5.13839343490042737674841580871, 6.45472420143965209062030297420, 7.03094791753694980624143862332, 7.58409654765888915337872268977, 8.577648669701098265484375790233

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