Properties

Label 2-570-1.1-c5-0-3
Degree $2$
Conductor $570$
Sign $1$
Analytic cond. $91.4187$
Root an. cond. $9.56131$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 9·3-s + 16·4-s − 25·5-s + 36·6-s − 10.5·7-s − 64·8-s + 81·9-s + 100·10-s + 66.1·11-s − 144·12-s − 874.·13-s + 42.0·14-s + 225·15-s + 256·16-s + 1.22e3·17-s − 324·18-s + 361·19-s − 400·20-s + 94.6·21-s − 264.·22-s − 3.27e3·23-s + 576·24-s + 625·25-s + 3.49e3·26-s − 729·27-s − 168.·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 − 0.0811·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.164·11-s − 0.288·12-s − 1.43·13-s + 0.0573·14-s + 0.258·15-s + 0.250·16-s + 1.03·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.0468·21-s − 0.116·22-s − 1.28·23-s + 0.204·24-s + 0.200·25-s + 1.01·26-s − 0.192·27-s − 0.0405·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(91.4187\)
Root analytic conductor: \(9.56131\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6084479929\)
\(L(\frac12)\) \(\approx\) \(0.6084479929\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
3 \( 1 + 9T \)
5 \( 1 + 25T \)
19 \( 1 - 361T \)
good7 \( 1 + 10.5T + 1.68e4T^{2} \)
11 \( 1 - 66.1T + 1.61e5T^{2} \)
13 \( 1 + 874.T + 3.71e5T^{2} \)
17 \( 1 - 1.22e3T + 1.41e6T^{2} \)
23 \( 1 + 3.27e3T + 6.43e6T^{2} \)
29 \( 1 + 5.08e3T + 2.05e7T^{2} \)
31 \( 1 - 4.22e3T + 2.86e7T^{2} \)
37 \( 1 + 1.93e3T + 6.93e7T^{2} \)
41 \( 1 - 1.27e4T + 1.15e8T^{2} \)
43 \( 1 - 2.47e3T + 1.47e8T^{2} \)
47 \( 1 + 1.73e4T + 2.29e8T^{2} \)
53 \( 1 + 2.02e3T + 4.18e8T^{2} \)
59 \( 1 + 3.59e4T + 7.14e8T^{2} \)
61 \( 1 - 1.19e3T + 8.44e8T^{2} \)
67 \( 1 + 1.23e4T + 1.35e9T^{2} \)
71 \( 1 - 1.34e3T + 1.80e9T^{2} \)
73 \( 1 + 8.59e4T + 2.07e9T^{2} \)
79 \( 1 + 2.98e4T + 3.07e9T^{2} \)
83 \( 1 - 1.03e5T + 3.93e9T^{2} \)
89 \( 1 - 6.96e4T + 5.58e9T^{2} \)
97 \( 1 - 7.98e3T + 8.58e9T^{2} \)
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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

−9.911158841976118861973721075362, −9.308838829658391643315891885351, −7.929366253079972514727553207492, −7.52498081836296588534832198466, −6.43981667502678933364910796415, −5.47200814550344151890154506355, −4.37262144756782869309085150541, −3.08317403865466998866041715502, −1.74707286055795662914597954084, −0.43353352370111677893852619581, 0.43353352370111677893852619581, 1.74707286055795662914597954084, 3.08317403865466998866041715502, 4.37262144756782869309085150541, 5.47200814550344151890154506355, 6.43981667502678933364910796415, 7.52498081836296588534832198466, 7.929366253079972514727553207492, 9.308838829658391643315891885351, 9.911158841976118861973721075362

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