Properties

Label 2-2475-1.1-c3-0-114
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $146.029$
Root an. cond. $12.0842$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.29·2-s − 6.31·4-s + 16.4·7-s + 18.5·8-s + 11·11-s + 4.28·13-s − 21.3·14-s + 26.3·16-s + 72.6·17-s + 133.·19-s − 14.2·22-s + 116.·23-s − 5.56·26-s − 103.·28-s + 144.·29-s + 235.·31-s − 182.·32-s − 94.4·34-s + 75.2·37-s − 173.·38-s + 355.·41-s − 2.86·43-s − 69.4·44-s − 151.·46-s − 83.0·47-s − 72.7·49-s − 27.0·52-s + ⋯
L(s)  = 1  − 0.459·2-s − 0.789·4-s + 0.887·7-s + 0.821·8-s + 0.301·11-s + 0.0914·13-s − 0.407·14-s + 0.411·16-s + 1.03·17-s + 1.61·19-s − 0.138·22-s + 1.05·23-s − 0.0419·26-s − 0.700·28-s + 0.926·29-s + 1.36·31-s − 1.01·32-s − 0.476·34-s + 0.334·37-s − 0.741·38-s + 1.35·41-s − 0.0101·43-s − 0.237·44-s − 0.486·46-s − 0.257·47-s − 0.212·49-s − 0.0721·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(146.029\)
Root analytic conductor: \(12.0842\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.199217080\)
\(L(\frac12)\) \(\approx\) \(2.199217080\)
\(L(\frac{5}{2})\) 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 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 + 1.29T + 8T^{2} \)
7 \( 1 - 16.4T + 343T^{2} \)
13 \( 1 - 4.28T + 2.19e3T^{2} \)
17 \( 1 - 72.6T + 4.91e3T^{2} \)
19 \( 1 - 133.T + 6.85e3T^{2} \)
23 \( 1 - 116.T + 1.21e4T^{2} \)
29 \( 1 - 144.T + 2.43e4T^{2} \)
31 \( 1 - 235.T + 2.97e4T^{2} \)
37 \( 1 - 75.2T + 5.06e4T^{2} \)
41 \( 1 - 355.T + 6.89e4T^{2} \)
43 \( 1 + 2.86T + 7.95e4T^{2} \)
47 \( 1 + 83.0T + 1.03e5T^{2} \)
53 \( 1 + 213.T + 1.48e5T^{2} \)
59 \( 1 + 477.T + 2.05e5T^{2} \)
61 \( 1 + 294.T + 2.26e5T^{2} \)
67 \( 1 - 1.00e3T + 3.00e5T^{2} \)
71 \( 1 - 1.02e3T + 3.57e5T^{2} \)
73 \( 1 + 76.8T + 3.89e5T^{2} \)
79 \( 1 - 456.T + 4.93e5T^{2} \)
83 \( 1 - 504.T + 5.71e5T^{2} \)
89 \( 1 + 849.T + 7.04e5T^{2} \)
97 \( 1 + 306.T + 9.12e5T^{2} \)
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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

−8.483624922772177755494920246042, −7.949427576968698478413117083670, −7.37063771263485509721495518652, −6.26281546019568426381877811306, −5.17217577996309341694055608666, −4.82720864746086950772120175988, −3.77698997167612417905100807348, −2.81991105523626563272006205786, −1.28831174734764235822355744127, −0.869762495617942438246275408521, 0.869762495617942438246275408521, 1.28831174734764235822355744127, 2.81991105523626563272006205786, 3.77698997167612417905100807348, 4.82720864746086950772120175988, 5.17217577996309341694055608666, 6.26281546019568426381877811306, 7.37063771263485509721495518652, 7.949427576968698478413117083670, 8.483624922772177755494920246042

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