Properties

Label 2-2475-1.1-c3-0-13
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  − 3.98·2-s + 7.89·4-s − 12.5·7-s + 0.411·8-s + 11·11-s − 36.0·13-s + 50.0·14-s − 64.8·16-s + 39.7·17-s − 148.·19-s − 43.8·22-s − 35.0·23-s + 143.·26-s − 99.2·28-s − 88.2·29-s − 166.·31-s + 255.·32-s − 158.·34-s − 85.2·37-s + 590.·38-s − 329.·41-s + 278.·43-s + 86.8·44-s + 139.·46-s − 272.·47-s − 185.·49-s − 284.·52-s + ⋯
L(s)  = 1  − 1.40·2-s + 0.987·4-s − 0.678·7-s + 0.0181·8-s + 0.301·11-s − 0.769·13-s + 0.956·14-s − 1.01·16-s + 0.566·17-s − 1.78·19-s − 0.425·22-s − 0.317·23-s + 1.08·26-s − 0.669·28-s − 0.565·29-s − 0.966·31-s + 1.40·32-s − 0.798·34-s − 0.378·37-s + 2.52·38-s − 1.25·41-s + 0.988·43-s + 0.297·44-s + 0.448·46-s − 0.846·47-s − 0.539·49-s − 0.759·52-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

\(L(2)\) \(\approx\) \(0.2864172553\)
\(L(\frac12)\) \(\approx\) \(0.2864172553\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.98T + 8T^{2} \)
7 \( 1 + 12.5T + 343T^{2} \)
13 \( 1 + 36.0T + 2.19e3T^{2} \)
17 \( 1 - 39.7T + 4.91e3T^{2} \)
19 \( 1 + 148.T + 6.85e3T^{2} \)
23 \( 1 + 35.0T + 1.21e4T^{2} \)
29 \( 1 + 88.2T + 2.43e4T^{2} \)
31 \( 1 + 166.T + 2.97e4T^{2} \)
37 \( 1 + 85.2T + 5.06e4T^{2} \)
41 \( 1 + 329.T + 6.89e4T^{2} \)
43 \( 1 - 278.T + 7.95e4T^{2} \)
47 \( 1 + 272.T + 1.03e5T^{2} \)
53 \( 1 - 223.T + 1.48e5T^{2} \)
59 \( 1 - 467.T + 2.05e5T^{2} \)
61 \( 1 - 752.T + 2.26e5T^{2} \)
67 \( 1 + 733.T + 3.00e5T^{2} \)
71 \( 1 + 537.T + 3.57e5T^{2} \)
73 \( 1 + 397.T + 3.89e5T^{2} \)
79 \( 1 + 1.07e3T + 4.93e5T^{2} \)
83 \( 1 - 683.T + 5.71e5T^{2} \)
89 \( 1 - 166.T + 7.04e5T^{2} \)
97 \( 1 - 694.T + 9.12e5T^{2} \)
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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

−8.723425438074035990666838971945, −7.974350488777853983566986568272, −7.17191769197918802988950665064, −6.62442966988940246820907820104, −5.67357320360548813919007695177, −4.55732724589626237305256342024, −3.62996731052640810240199669027, −2.41154825904253618367974644540, −1.59616287416682399006285397730, −0.29063864191168474430159525186, 0.29063864191168474430159525186, 1.59616287416682399006285397730, 2.41154825904253618367974644540, 3.62996731052640810240199669027, 4.55732724589626237305256342024, 5.67357320360548813919007695177, 6.62442966988940246820907820104, 7.17191769197918802988950665064, 7.974350488777853983566986568272, 8.723425438074035990666838971945

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