Properties

Label 2-560-1.1-c1-0-7
Degree $2$
Conductor $560$
Sign $1$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + 2.56·3-s + 5-s + 7-s + 3.56·9-s − 2.56·11-s + 4.56·13-s + 2.56·15-s − 4.56·17-s − 1.12·19-s + 2.56·21-s + 5.12·23-s + 25-s + 1.43·27-s − 5.68·29-s − 6.56·33-s + 35-s + 6·37-s + 11.6·39-s − 3.12·41-s − 9.12·43-s + 3.56·45-s − 3.68·47-s + 49-s − 11.6·51-s + 3.12·53-s − 2.56·55-s − 2.87·57-s + ⋯
L(s)  = 1  + 1.47·3-s + 0.447·5-s + 0.377·7-s + 1.18·9-s − 0.772·11-s + 1.26·13-s + 0.661·15-s − 1.10·17-s − 0.257·19-s + 0.558·21-s + 1.06·23-s + 0.200·25-s + 0.276·27-s − 1.05·29-s − 1.14·33-s + 0.169·35-s + 0.986·37-s + 1.87·39-s − 0.487·41-s − 1.39·43-s + 0.530·45-s − 0.537·47-s + 0.142·49-s − 1.63·51-s + 0.428·53-s − 0.345·55-s − 0.381·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.525502063\)
\(L(\frac12)\) \(\approx\) \(2.525502063\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
good3 \( 1 - 2.56T + 3T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 + 4.56T + 17T^{2} \)
19 \( 1 + 1.12T + 19T^{2} \)
23 \( 1 - 5.12T + 23T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 3.12T + 41T^{2} \)
43 \( 1 + 9.12T + 43T^{2} \)
47 \( 1 + 3.68T + 47T^{2} \)
53 \( 1 - 3.12T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 9.36T + 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 - 6.56T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 7.12T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−10.71852278879301063064640206695, −9.665051256331339298262348336268, −8.827835147304446976532813852039, −8.351613035344276371663628470056, −7.40369311082293681903432731110, −6.31541053035408999850880288606, −5.04595920059395037042341789054, −3.84123208348491185714193855596, −2.78921232307861910493038679196, −1.74497648316645092426880643265, 1.74497648316645092426880643265, 2.78921232307861910493038679196, 3.84123208348491185714193855596, 5.04595920059395037042341789054, 6.31541053035408999850880288606, 7.40369311082293681903432731110, 8.351613035344276371663628470056, 8.827835147304446976532813852039, 9.665051256331339298262348336268, 10.71852278879301063064640206695

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