Properties

Label 2-570-1.1-c5-0-4
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 − 230.·7-s − 64·8-s + 81·9-s + 100·10-s + 628.·11-s − 144·12-s + 803.·13-s + 922.·14-s + 225·15-s + 256·16-s − 802.·17-s − 324·18-s + 361·19-s − 400·20-s + 2.07e3·21-s − 2.51e3·22-s − 2.59e3·23-s + 576·24-s + 625·25-s − 3.21e3·26-s − 729·27-s − 3.69e3·28-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 + 1.56·11-s − 0.288·12-s + 1.31·13-s + 1.25·14-s + 0.258·15-s + 0.250·16-s − 0.673·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 1.02·21-s − 1.10·22-s − 1.02·23-s + 0.204·24-s + 0.200·25-s − 0.932·26-s − 0.192·27-s − 0.889·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5920156993\)
\(L(\frac12)\) \(\approx\) \(0.5920156993\)
\(L(\frac{7}{2})\) 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 \( 1 + 230.T + 1.68e4T^{2} \)
11 \( 1 - 628.T + 1.61e5T^{2} \)
13 \( 1 - 803.T + 3.71e5T^{2} \)
17 \( 1 + 802.T + 1.41e6T^{2} \)
23 \( 1 + 2.59e3T + 6.43e6T^{2} \)
29 \( 1 + 1.93e3T + 2.05e7T^{2} \)
31 \( 1 + 4.07e3T + 2.86e7T^{2} \)
37 \( 1 + 8.71e3T + 6.93e7T^{2} \)
41 \( 1 + 1.15e4T + 1.15e8T^{2} \)
43 \( 1 + 8.07e3T + 1.47e8T^{2} \)
47 \( 1 - 3.31e3T + 2.29e8T^{2} \)
53 \( 1 + 1.80e4T + 4.18e8T^{2} \)
59 \( 1 - 1.31e4T + 7.14e8T^{2} \)
61 \( 1 - 1.48e3T + 8.44e8T^{2} \)
67 \( 1 - 6.70e4T + 1.35e9T^{2} \)
71 \( 1 - 2.15e3T + 1.80e9T^{2} \)
73 \( 1 + 3.66e4T + 2.07e9T^{2} \)
79 \( 1 + 9.94e3T + 3.07e9T^{2} \)
83 \( 1 + 3.98e4T + 3.93e9T^{2} \)
89 \( 1 - 1.31e5T + 5.58e9T^{2} \)
97 \( 1 + 1.07e5T + 8.58e9T^{2} \)
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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

−9.841693010553907654034559691786, −9.185225257437353599303938031500, −8.400530401296270714192716436250, −6.93602464989096327073657692022, −6.58156873357932077037621243691, −5.76302336806073178306341762779, −3.95771650768883332522073245765, −3.40291540628465690194466327935, −1.64980369824410100090104980436, −0.43808037305316715281099766231, 0.43808037305316715281099766231, 1.64980369824410100090104980436, 3.40291540628465690194466327935, 3.95771650768883332522073245765, 5.76302336806073178306341762779, 6.58156873357932077037621243691, 6.93602464989096327073657692022, 8.400530401296270714192716436250, 9.185225257437353599303938031500, 9.841693010553907654034559691786

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