Properties

Label 2-570-1.1-c5-0-14
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 − 34.2·7-s − 64·8-s + 81·9-s + 100·10-s + 148.·11-s + 144·12-s + 1.14e3·13-s + 137.·14-s − 225·15-s + 256·16-s + 1.86e3·17-s − 324·18-s − 361·19-s − 400·20-s − 308.·21-s − 595.·22-s + 356.·23-s − 576·24-s + 625·25-s − 4.56e3·26-s + 729·27-s − 548.·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 − 0.264·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.370·11-s + 0.288·12-s + 1.87·13-s + 0.187·14-s − 0.258·15-s + 0.250·16-s + 1.56·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.152·21-s − 0.262·22-s + 0.140·23-s − 0.204·24-s + 0.200·25-s − 1.32·26-s + 0.192·27-s − 0.132·28-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

\(L(3)\) \(\approx\) \(2.008570321\)
\(L(\frac12)\) \(\approx\) \(2.008570321\)
\(L(\frac{7}{2})\) 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 \( 1 + 34.2T + 1.68e4T^{2} \)
11 \( 1 - 148.T + 1.61e5T^{2} \)
13 \( 1 - 1.14e3T + 3.71e5T^{2} \)
17 \( 1 - 1.86e3T + 1.41e6T^{2} \)
23 \( 1 - 356.T + 6.43e6T^{2} \)
29 \( 1 + 5.82e3T + 2.05e7T^{2} \)
31 \( 1 + 1.04e4T + 2.86e7T^{2} \)
37 \( 1 - 6.39e3T + 6.93e7T^{2} \)
41 \( 1 - 3.87e3T + 1.15e8T^{2} \)
43 \( 1 - 2.31e4T + 1.47e8T^{2} \)
47 \( 1 + 2.41e4T + 2.29e8T^{2} \)
53 \( 1 + 2.47e4T + 4.18e8T^{2} \)
59 \( 1 - 2.92e4T + 7.14e8T^{2} \)
61 \( 1 - 559.T + 8.44e8T^{2} \)
67 \( 1 - 3.46e4T + 1.35e9T^{2} \)
71 \( 1 - 6.39e4T + 1.80e9T^{2} \)
73 \( 1 - 1.58e4T + 2.07e9T^{2} \)
79 \( 1 - 7.23e4T + 3.07e9T^{2} \)
83 \( 1 + 7.01e4T + 3.93e9T^{2} \)
89 \( 1 - 7.24e4T + 5.58e9T^{2} \)
97 \( 1 - 7.22e3T + 8.58e9T^{2} \)
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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

−9.674713067437949316982864316237, −9.124113544872450169392672744122, −8.172744123865122778553171911120, −7.59749518203413614836645327525, −6.49991243858038671613734097158, −5.56916812235438299134443442831, −3.83926148696020834458429942717, −3.32947817198922500001471701669, −1.77852955548984555289461001209, −0.78702533463434135601803253129, 0.78702533463434135601803253129, 1.77852955548984555289461001209, 3.32947817198922500001471701669, 3.83926148696020834458429942717, 5.56916812235438299134443442831, 6.49991243858038671613734097158, 7.59749518203413614836645327525, 8.172744123865122778553171911120, 9.124113544872450169392672744122, 9.674713067437949316982864316237

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