Properties

Label 2-570-1.1-c5-0-2
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 − 188.·7-s − 64·8-s + 81·9-s + 100·10-s + 163.·11-s + 144·12-s − 1.06e3·13-s + 753.·14-s − 225·15-s + 256·16-s + 726.·17-s − 324·18-s − 361·19-s − 400·20-s − 1.69e3·21-s − 653.·22-s − 1.89e3·23-s − 576·24-s + 625·25-s + 4.27e3·26-s + 729·27-s − 3.01e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 1.45·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.406·11-s + 0.288·12-s − 1.75·13-s + 1.02·14-s − 0.258·15-s + 0.250·16-s + 0.609·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.839·21-s − 0.287·22-s − 0.747·23-s − 0.204·24-s + 0.200·25-s + 1.23·26-s + 0.192·27-s − 0.726·28-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

\(L(3)\) \(\approx\) \(0.7331863320\)
\(L(\frac12)\) \(\approx\) \(0.7331863320\)
\(L(\frac{7}{2})\) 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 \( 1 + 188.T + 1.68e4T^{2} \)
11 \( 1 - 163.T + 1.61e5T^{2} \)
13 \( 1 + 1.06e3T + 3.71e5T^{2} \)
17 \( 1 - 726.T + 1.41e6T^{2} \)
23 \( 1 + 1.89e3T + 6.43e6T^{2} \)
29 \( 1 + 5.08e3T + 2.05e7T^{2} \)
31 \( 1 - 1.14e3T + 2.86e7T^{2} \)
37 \( 1 + 1.38e4T + 6.93e7T^{2} \)
41 \( 1 - 7.30e3T + 1.15e8T^{2} \)
43 \( 1 + 7.62e3T + 1.47e8T^{2} \)
47 \( 1 + 2.64e3T + 2.29e8T^{2} \)
53 \( 1 - 2.37e4T + 4.18e8T^{2} \)
59 \( 1 - 2.56e4T + 7.14e8T^{2} \)
61 \( 1 - 3.02e4T + 8.44e8T^{2} \)
67 \( 1 + 1.67e4T + 1.35e9T^{2} \)
71 \( 1 + 1.87e4T + 1.80e9T^{2} \)
73 \( 1 - 3.41e4T + 2.07e9T^{2} \)
79 \( 1 - 7.09e4T + 3.07e9T^{2} \)
83 \( 1 + 7.98e4T + 3.93e9T^{2} \)
89 \( 1 + 1.97e4T + 5.58e9T^{2} \)
97 \( 1 - 7.69e4T + 8.58e9T^{2} \)
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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

−9.803197973601564448241204778256, −9.245106913359455615317075995999, −8.236026379995925718311011346812, −7.28536470246558265996210228849, −6.75703182645183434365380445574, −5.47980017227961197502989770805, −3.97711968628989544364247143225, −3.08680486680499968139836628227, −2.06563170752911657283740196605, −0.42807622797171266270546058512, 0.42807622797171266270546058512, 2.06563170752911657283740196605, 3.08680486680499968139836628227, 3.97711968628989544364247143225, 5.47980017227961197502989770805, 6.75703182645183434365380445574, 7.28536470246558265996210228849, 8.236026379995925718311011346812, 9.245106913359455615317075995999, 9.803197973601564448241204778256

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