Properties

Label 2-570-1.1-c5-0-33
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 − 171.·7-s − 64·8-s + 81·9-s − 100·10-s + 409.·11-s − 144·12-s − 730.·13-s + 686.·14-s − 225·15-s + 256·16-s + 248.·17-s − 324·18-s + 361·19-s + 400·20-s + 1.54e3·21-s − 1.63e3·22-s + 1.71e3·23-s + 576·24-s + 625·25-s + 2.92e3·26-s − 729·27-s − 2.74e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.32·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.02·11-s − 0.288·12-s − 1.19·13-s + 0.935·14-s − 0.258·15-s + 0.250·16-s + 0.208·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.764·21-s − 0.721·22-s + 0.675·23-s + 0.204·24-s + 0.200·25-s + 0.847·26-s − 0.192·27-s − 0.661·28-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 \( 1 + 171.T + 1.68e4T^{2} \)
11 \( 1 - 409.T + 1.61e5T^{2} \)
13 \( 1 + 730.T + 3.71e5T^{2} \)
17 \( 1 - 248.T + 1.41e6T^{2} \)
23 \( 1 - 1.71e3T + 6.43e6T^{2} \)
29 \( 1 + 935.T + 2.05e7T^{2} \)
31 \( 1 - 396.T + 2.86e7T^{2} \)
37 \( 1 + 495.T + 6.93e7T^{2} \)
41 \( 1 - 1.71e4T + 1.15e8T^{2} \)
43 \( 1 + 2.20e4T + 1.47e8T^{2} \)
47 \( 1 - 1.18e4T + 2.29e8T^{2} \)
53 \( 1 - 6.64e3T + 4.18e8T^{2} \)
59 \( 1 - 606.T + 7.14e8T^{2} \)
61 \( 1 + 3.38e4T + 8.44e8T^{2} \)
67 \( 1 - 4.10e3T + 1.35e9T^{2} \)
71 \( 1 - 3.96e4T + 1.80e9T^{2} \)
73 \( 1 - 4.05e4T + 2.07e9T^{2} \)
79 \( 1 - 6.26e4T + 3.07e9T^{2} \)
83 \( 1 + 3.13e4T + 3.93e9T^{2} \)
89 \( 1 - 8.42e4T + 5.58e9T^{2} \)
97 \( 1 - 1.75e4T + 8.58e9T^{2} \)
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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

−9.576909880779972568707492993685, −9.005405157706773738831385633452, −7.56272577173055801211874233334, −6.74921631400134490120534337524, −6.14790344756453748547629432099, −5.03267713282425962922143212345, −3.61009421495241861517796178312, −2.45802004752113121626778136770, −1.08676122400240349633269842782, 0, 1.08676122400240349633269842782, 2.45802004752113121626778136770, 3.61009421495241861517796178312, 5.03267713282425962922143212345, 6.14790344756453748547629432099, 6.74921631400134490120534337524, 7.56272577173055801211874233334, 9.005405157706773738831385633452, 9.576909880779972568707492993685

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