Properties

Label 2-570-1.1-c5-0-47
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 $1$

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 + 185.·7-s − 64·8-s + 81·9-s − 100·10-s + 132.·11-s − 144·12-s + 637.·13-s − 740.·14-s − 225·15-s + 256·16-s − 2.26e3·17-s − 324·18-s + 361·19-s + 400·20-s − 1.66e3·21-s − 531.·22-s − 1.23e3·23-s + 576·24-s + 625·25-s − 2.55e3·26-s − 729·27-s + 2.96e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.42·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.331·11-s − 0.288·12-s + 1.04·13-s − 1.01·14-s − 0.258·15-s + 0.250·16-s − 1.90·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.824·21-s − 0.234·22-s − 0.485·23-s + 0.204·24-s + 0.200·25-s − 0.739·26-s − 0.192·27-s + 0.714·28-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: \(1\)
Selberg data: \((2,\ 570,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 185.T + 1.68e4T^{2} \)
11 \( 1 - 132.T + 1.61e5T^{2} \)
13 \( 1 - 637.T + 3.71e5T^{2} \)
17 \( 1 + 2.26e3T + 1.41e6T^{2} \)
23 \( 1 + 1.23e3T + 6.43e6T^{2} \)
29 \( 1 + 6.56e3T + 2.05e7T^{2} \)
31 \( 1 + 9.17e3T + 2.86e7T^{2} \)
37 \( 1 - 8.68e3T + 6.93e7T^{2} \)
41 \( 1 + 4.87e3T + 1.15e8T^{2} \)
43 \( 1 + 1.15e4T + 1.47e8T^{2} \)
47 \( 1 + 1.54e4T + 2.29e8T^{2} \)
53 \( 1 + 2.49e3T + 4.18e8T^{2} \)
59 \( 1 - 1.41e4T + 7.14e8T^{2} \)
61 \( 1 + 1.88e4T + 8.44e8T^{2} \)
67 \( 1 - 1.96e3T + 1.35e9T^{2} \)
71 \( 1 - 3.19e4T + 1.80e9T^{2} \)
73 \( 1 + 6.41e4T + 2.07e9T^{2} \)
79 \( 1 - 8.81e4T + 3.07e9T^{2} \)
83 \( 1 + 1.28e4T + 3.93e9T^{2} \)
89 \( 1 + 6.01e4T + 5.58e9T^{2} \)
97 \( 1 + 3.38e4T + 8.58e9T^{2} \)
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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

−9.397150897048917357045436843282, −8.717667519189554505861011003619, −7.83517805453089345855286481857, −6.82492015931688738054624930702, −5.95007657043275928242283583279, −4.97874486878879759053027827247, −3.87688193900449413936159254413, −2.03478606288426683579999800418, −1.43225706458031069663282292998, 0, 1.43225706458031069663282292998, 2.03478606288426683579999800418, 3.87688193900449413936159254413, 4.97874486878879759053027827247, 5.95007657043275928242283583279, 6.82492015931688738054624930702, 7.83517805453089345855286481857, 8.717667519189554505861011003619, 9.397150897048917357045436843282

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