Properties

Label 2-570-1.1-c5-0-30
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 + 229.·7-s − 64·8-s + 81·9-s − 100·10-s − 291.·11-s + 144·12-s + 1.15e3·13-s − 919.·14-s + 225·15-s + 256·16-s + 1.16e3·17-s − 324·18-s + 361·19-s + 400·20-s + 2.06e3·21-s + 1.16e3·22-s − 3.80e3·23-s − 576·24-s + 625·25-s − 4.63e3·26-s + 729·27-s + 3.67e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.77·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.725·11-s + 0.288·12-s + 1.89·13-s − 1.25·14-s + 0.258·15-s + 0.250·16-s + 0.975·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 1.02·21-s + 0.512·22-s − 1.50·23-s − 0.204·24-s + 0.200·25-s − 1.34·26-s + 0.192·27-s + 0.886·28-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

\(L(3)\) \(\approx\) \(3.210742113\)
\(L(\frac12)\) \(\approx\) \(3.210742113\)
\(L(\frac{7}{2})\) 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 \( 1 - 229.T + 1.68e4T^{2} \)
11 \( 1 + 291.T + 1.61e5T^{2} \)
13 \( 1 - 1.15e3T + 3.71e5T^{2} \)
17 \( 1 - 1.16e3T + 1.41e6T^{2} \)
23 \( 1 + 3.80e3T + 6.43e6T^{2} \)
29 \( 1 - 7.36e3T + 2.05e7T^{2} \)
31 \( 1 - 6.14e3T + 2.86e7T^{2} \)
37 \( 1 + 1.72e3T + 6.93e7T^{2} \)
41 \( 1 + 1.10e4T + 1.15e8T^{2} \)
43 \( 1 - 1.68e4T + 1.47e8T^{2} \)
47 \( 1 + 6.74e3T + 2.29e8T^{2} \)
53 \( 1 + 3.85e4T + 4.18e8T^{2} \)
59 \( 1 - 3.09e4T + 7.14e8T^{2} \)
61 \( 1 + 7.28e3T + 8.44e8T^{2} \)
67 \( 1 + 2.56e4T + 1.35e9T^{2} \)
71 \( 1 + 7.32e3T + 1.80e9T^{2} \)
73 \( 1 + 8.11e4T + 2.07e9T^{2} \)
79 \( 1 + 1.78e3T + 3.07e9T^{2} \)
83 \( 1 - 4.14e4T + 3.93e9T^{2} \)
89 \( 1 - 5.20e4T + 5.58e9T^{2} \)
97 \( 1 + 9.15e4T + 8.58e9T^{2} \)
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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

−10.06432391028400221728845347383, −8.815408916341289331041944040276, −8.142021917787282396233894005400, −7.84432285112648576979354763346, −6.39117350858204798412437045619, −5.43867897393753459748578897190, −4.27711143770156740478370330081, −2.93006488273541922985060124379, −1.72165015650152635579501601491, −1.07597534151201279847204857993, 1.07597534151201279847204857993, 1.72165015650152635579501601491, 2.93006488273541922985060124379, 4.27711143770156740478370330081, 5.43867897393753459748578897190, 6.39117350858204798412437045619, 7.84432285112648576979354763346, 8.142021917787282396233894005400, 8.815408916341289331041944040276, 10.06432391028400221728845347383

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