Properties

Label 2-570-1.1-c5-0-20
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 − 88.0·7-s − 64·8-s + 81·9-s − 100·10-s + 673.·11-s + 144·12-s + 382.·13-s + 352.·14-s + 225·15-s + 256·16-s + 1.55e3·17-s − 324·18-s + 361·19-s + 400·20-s − 792.·21-s − 2.69e3·22-s − 893.·23-s − 576·24-s + 625·25-s − 1.52e3·26-s + 729·27-s − 1.40e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.679·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.67·11-s + 0.288·12-s + 0.626·13-s + 0.480·14-s + 0.258·15-s + 0.250·16-s + 1.30·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.392·21-s − 1.18·22-s − 0.352·23-s − 0.204·24-s + 0.200·25-s − 0.443·26-s + 0.192·27-s − 0.339·28-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: $\chi_{570} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.489457863\)
\(L(\frac12)\) \(\approx\) \(2.489457863\)
\(L(\frac{7}{2})\) 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 \( 1 + 88.0T + 1.68e4T^{2} \)
11 \( 1 - 673.T + 1.61e5T^{2} \)
13 \( 1 - 382.T + 3.71e5T^{2} \)
17 \( 1 - 1.55e3T + 1.41e6T^{2} \)
23 \( 1 + 893.T + 6.43e6T^{2} \)
29 \( 1 + 474.T + 2.05e7T^{2} \)
31 \( 1 + 2.42e3T + 2.86e7T^{2} \)
37 \( 1 + 4.39e3T + 6.93e7T^{2} \)
41 \( 1 - 249.T + 1.15e8T^{2} \)
43 \( 1 + 2.05e3T + 1.47e8T^{2} \)
47 \( 1 - 1.57e4T + 2.29e8T^{2} \)
53 \( 1 - 1.41e4T + 4.18e8T^{2} \)
59 \( 1 - 2.02e4T + 7.14e8T^{2} \)
61 \( 1 + 5.37e3T + 8.44e8T^{2} \)
67 \( 1 - 2.63e4T + 1.35e9T^{2} \)
71 \( 1 + 6.15e4T + 1.80e9T^{2} \)
73 \( 1 - 1.36e4T + 2.07e9T^{2} \)
79 \( 1 + 2.25e4T + 3.07e9T^{2} \)
83 \( 1 - 6.00e4T + 3.93e9T^{2} \)
89 \( 1 + 5.85e4T + 5.58e9T^{2} \)
97 \( 1 - 1.00e5T + 8.58e9T^{2} \)
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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

−9.720310621953929151428995911183, −9.181065105725501362441464940654, −8.412429070745202052139722810021, −7.31144338047617505955717758667, −6.50453214492779801457285343888, −5.65554553337824510103351427267, −3.93840482561963732131821276935, −3.16693787605141757273641782859, −1.78310336414862252472290422783, −0.891955783496050536116547079658, 0.891955783496050536116547079658, 1.78310336414862252472290422783, 3.16693787605141757273641782859, 3.93840482561963732131821276935, 5.65554553337824510103351427267, 6.50453214492779801457285343888, 7.31144338047617505955717758667, 8.412429070745202052139722810021, 9.181065105725501362441464940654, 9.720310621953929151428995911183

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