Properties

Label 2-570-1.1-c7-0-60
Degree $2$
Conductor $570$
Sign $-1$
Analytic cond. $178.059$
Root an. cond. $13.3438$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 27·3-s + 64·4-s + 125·5-s − 216·6-s − 1.16e3·7-s + 512·8-s + 729·9-s + 1.00e3·10-s − 144.·11-s − 1.72e3·12-s − 3.04e3·13-s − 9.32e3·14-s − 3.37e3·15-s + 4.09e3·16-s + 6.70e3·17-s + 5.83e3·18-s + 6.85e3·19-s + 8.00e3·20-s + 3.14e4·21-s − 1.15e3·22-s + 1.54e4·23-s − 1.38e4·24-s + 1.56e4·25-s − 2.43e4·26-s − 1.96e4·27-s − 7.46e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.28·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.0326·11-s − 0.288·12-s − 0.384·13-s − 0.908·14-s − 0.258·15-s + 0.250·16-s + 0.330·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.741·21-s − 0.0230·22-s + 0.264·23-s − 0.204·24-s + 0.199·25-s − 0.272·26-s − 0.192·27-s − 0.642·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(178.059\)
Root analytic conductor: \(13.3438\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 570,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8T \)
3 \( 1 + 27T \)
5 \( 1 - 125T \)
19 \( 1 - 6.85e3T \)
good7 \( 1 + 1.16e3T + 8.23e5T^{2} \)
11 \( 1 + 144.T + 1.94e7T^{2} \)
13 \( 1 + 3.04e3T + 6.27e7T^{2} \)
17 \( 1 - 6.70e3T + 4.10e8T^{2} \)
23 \( 1 - 1.54e4T + 3.40e9T^{2} \)
29 \( 1 - 1.80e5T + 1.72e10T^{2} \)
31 \( 1 - 1.73e5T + 2.75e10T^{2} \)
37 \( 1 + 4.52e5T + 9.49e10T^{2} \)
41 \( 1 - 3.38e5T + 1.94e11T^{2} \)
43 \( 1 + 3.57e5T + 2.71e11T^{2} \)
47 \( 1 + 4.86e5T + 5.06e11T^{2} \)
53 \( 1 + 1.80e6T + 1.17e12T^{2} \)
59 \( 1 - 1.20e6T + 2.48e12T^{2} \)
61 \( 1 + 6.14e5T + 3.14e12T^{2} \)
67 \( 1 - 2.28e6T + 6.06e12T^{2} \)
71 \( 1 - 2.30e6T + 9.09e12T^{2} \)
73 \( 1 + 1.41e6T + 1.10e13T^{2} \)
79 \( 1 + 5.82e6T + 1.92e13T^{2} \)
83 \( 1 - 5.65e6T + 2.71e13T^{2} \)
89 \( 1 - 5.01e6T + 4.42e13T^{2} \)
97 \( 1 + 5.52e6T + 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.510285057239034544370821287062, −8.181786172098699667234117936981, −6.87424415374293487561554112058, −6.46226604641903186924078677140, −5.50362967236820008764742892772, −4.66802448040047593795174210712, −3.43757940186406572271588510180, −2.61672442849577308560504444531, −1.22754601498258425998302544240, 0, 1.22754601498258425998302544240, 2.61672442849577308560504444531, 3.43757940186406572271588510180, 4.66802448040047593795174210712, 5.50362967236820008764742892772, 6.46226604641903186924078677140, 6.87424415374293487561554112058, 8.181786172098699667234117936981, 9.510285057239034544370821287062

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