Properties

Label 2-574-1.1-c1-0-15
Degree $2$
Conductor $574$
Sign $1$
Analytic cond. $4.58341$
Root an. cond. $2.14089$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 2.73·5-s + 2·6-s + 7-s + 8-s + 9-s + 2.73·10-s − 2.73·11-s + 2·12-s − 5.46·13-s + 14-s + 5.46·15-s + 16-s + 18-s − 3.46·19-s + 2.73·20-s + 2·21-s − 2.73·22-s − 2·23-s + 2·24-s + 2.46·25-s − 5.46·26-s − 4·27-s + 28-s − 7.66·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 0.5·4-s + 1.22·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.863·10-s − 0.823·11-s + 0.577·12-s − 1.51·13-s + 0.267·14-s + 1.41·15-s + 0.250·16-s + 0.235·18-s − 0.794·19-s + 0.610·20-s + 0.436·21-s − 0.582·22-s − 0.417·23-s + 0.408·24-s + 0.492·25-s − 1.07·26-s − 0.769·27-s + 0.188·28-s − 1.42·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(574\)    =    \(2 \cdot 7 \cdot 41\)
Sign: $1$
Analytic conductor: \(4.58341\)
Root analytic conductor: \(2.14089\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 574,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.460168420\)
\(L(\frac12)\) \(\approx\) \(3.460168420\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
41 \( 1 - T \)
good3 \( 1 - 2T + 3T^{2} \)
5 \( 1 - 2.73T + 5T^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
13 \( 1 + 5.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + 7.66T + 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 - 4.92T + 37T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 - 0.535T + 47T^{2} \)
53 \( 1 - 7.66T + 53T^{2} \)
59 \( 1 - 13.1T + 59T^{2} \)
61 \( 1 - 6.73T + 61T^{2} \)
67 \( 1 - 0.196T + 67T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + 8.92T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4.73T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.57153349817273593405619244112, −9.813132539758024597169888458094, −9.066189793075234107295096885690, −7.967017025517771098830697611375, −7.26569317579054567255770987474, −5.93172699425919030021950594286, −5.19300327966827695169554689757, −4.01036689106736926473861118481, −2.47520088288264753241437176258, −2.23595805020013558835760984666, 2.23595805020013558835760984666, 2.47520088288264753241437176258, 4.01036689106736926473861118481, 5.19300327966827695169554689757, 5.93172699425919030021950594286, 7.26569317579054567255770987474, 7.967017025517771098830697611375, 9.066189793075234107295096885690, 9.813132539758024597169888458094, 10.57153349817273593405619244112

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