Properties

Label 2-570-1.1-c5-0-28
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 − 177.·7-s − 64·8-s + 81·9-s − 100·10-s − 464.·11-s − 144·12-s + 535.·13-s + 709.·14-s − 225·15-s + 256·16-s + 173.·17-s − 324·18-s + 361·19-s + 400·20-s + 1.59e3·21-s + 1.85e3·22-s + 2.05e3·23-s + 576·24-s + 625·25-s − 2.14e3·26-s − 729·27-s − 2.83e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.36·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.15·11-s − 0.288·12-s + 0.878·13-s + 0.967·14-s − 0.258·15-s + 0.250·16-s + 0.145·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.789·21-s + 0.818·22-s + 0.808·23-s + 0.204·24-s + 0.200·25-s − 0.621·26-s − 0.192·27-s − 0.684·28-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 \( 1 + 177.T + 1.68e4T^{2} \)
11 \( 1 + 464.T + 1.61e5T^{2} \)
13 \( 1 - 535.T + 3.71e5T^{2} \)
17 \( 1 - 173.T + 1.41e6T^{2} \)
23 \( 1 - 2.05e3T + 6.43e6T^{2} \)
29 \( 1 - 6.00e3T + 2.05e7T^{2} \)
31 \( 1 + 7.40e3T + 2.86e7T^{2} \)
37 \( 1 + 1.70e3T + 6.93e7T^{2} \)
41 \( 1 + 731.T + 1.15e8T^{2} \)
43 \( 1 - 2.41e4T + 1.47e8T^{2} \)
47 \( 1 + 2.28e4T + 2.29e8T^{2} \)
53 \( 1 + 5.42e3T + 4.18e8T^{2} \)
59 \( 1 - 1.67e4T + 7.14e8T^{2} \)
61 \( 1 - 5.10e4T + 8.44e8T^{2} \)
67 \( 1 + 5.94e4T + 1.35e9T^{2} \)
71 \( 1 - 5.79e3T + 1.80e9T^{2} \)
73 \( 1 + 5.63e4T + 2.07e9T^{2} \)
79 \( 1 - 7.19e4T + 3.07e9T^{2} \)
83 \( 1 - 8.36e4T + 3.93e9T^{2} \)
89 \( 1 - 1.25e5T + 5.58e9T^{2} \)
97 \( 1 - 1.97e4T + 8.58e9T^{2} \)
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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

−9.583915497322751019796514302348, −8.837603862075191261124988078627, −7.69211509486292194487593902968, −6.73375260143736089650205127208, −6.03754677328473983322164165738, −5.14266377876641583340562290282, −3.52125773833281939904756082588, −2.53094550363241591353237435252, −1.04917506594246080307402499108, 0, 1.04917506594246080307402499108, 2.53094550363241591353237435252, 3.52125773833281939904756082588, 5.14266377876641583340562290282, 6.03754677328473983322164165738, 6.73375260143736089650205127208, 7.69211509486292194487593902968, 8.837603862075191261124988078627, 9.583915497322751019796514302348

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