Properties

Label 2-731-1.1-c1-0-49
Degree $2$
Conductor $731$
Sign $1$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
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  + 2.29·2-s + 0.857·3-s + 3.25·4-s + 2.44·5-s + 1.96·6-s + 2.38·7-s + 2.86·8-s − 2.26·9-s + 5.60·10-s − 0.0820·11-s + 2.79·12-s − 5.52·13-s + 5.45·14-s + 2.09·15-s + 0.0710·16-s − 17-s − 5.18·18-s − 7.10·19-s + 7.94·20-s + 2.04·21-s − 0.187·22-s + 7.75·23-s + 2.46·24-s + 0.976·25-s − 12.6·26-s − 4.51·27-s + 7.73·28-s + ⋯
L(s)  = 1  + 1.62·2-s + 0.495·3-s + 1.62·4-s + 1.09·5-s + 0.802·6-s + 0.899·7-s + 1.01·8-s − 0.754·9-s + 1.77·10-s − 0.0247·11-s + 0.805·12-s − 1.53·13-s + 1.45·14-s + 0.541·15-s + 0.0177·16-s − 0.242·17-s − 1.22·18-s − 1.62·19-s + 1.77·20-s + 0.445·21-s − 0.0400·22-s + 1.61·23-s + 0.502·24-s + 0.195·25-s − 2.48·26-s − 0.869·27-s + 1.46·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $1$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.728925299\)
\(L(\frac12)\) \(\approx\) \(4.728925299\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + T \)
43 \( 1 - T \)
good2 \( 1 - 2.29T + 2T^{2} \)
3 \( 1 - 0.857T + 3T^{2} \)
5 \( 1 - 2.44T + 5T^{2} \)
7 \( 1 - 2.38T + 7T^{2} \)
11 \( 1 + 0.0820T + 11T^{2} \)
13 \( 1 + 5.52T + 13T^{2} \)
19 \( 1 + 7.10T + 19T^{2} \)
23 \( 1 - 7.75T + 23T^{2} \)
29 \( 1 - 1.83T + 29T^{2} \)
31 \( 1 + 7.18T + 31T^{2} \)
37 \( 1 - 2.70T + 37T^{2} \)
41 \( 1 - 9.68T + 41T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 2.50T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 3.96T + 61T^{2} \)
67 \( 1 - 16.2T + 67T^{2} \)
71 \( 1 + 9.09T + 71T^{2} \)
73 \( 1 - 7.34T + 73T^{2} \)
79 \( 1 - 2.80T + 79T^{2} \)
83 \( 1 - 7.03T + 83T^{2} \)
89 \( 1 + 1.65T + 89T^{2} \)
97 \( 1 - 3.97T + 97T^{2} \)
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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

−10.76076169143602978524415725616, −9.471041264415131328392183953476, −8.742384470041169098269343183630, −7.54038199690140333309099146487, −6.58326486081370258332264853162, −5.58771512020994076745777399030, −5.03593325959111724128072087825, −4.06661037176460602157145990021, −2.59017130177178500859680446425, −2.20206470175289712898790855802, 2.20206470175289712898790855802, 2.59017130177178500859680446425, 4.06661037176460602157145990021, 5.03593325959111724128072087825, 5.58771512020994076745777399030, 6.58326486081370258332264853162, 7.54038199690140333309099146487, 8.742384470041169098269343183630, 9.471041264415131328392183953476, 10.76076169143602978524415725616

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