Properties

Label 2-731-1.1-c1-0-4
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  + 0.169·2-s − 1.63·3-s − 1.97·4-s − 3.37·5-s − 0.277·6-s + 0.890·7-s − 0.675·8-s − 0.335·9-s − 0.574·10-s − 2.91·11-s + 3.21·12-s − 4.89·13-s + 0.151·14-s + 5.51·15-s + 3.82·16-s − 17-s − 0.0571·18-s + 7.26·19-s + 6.65·20-s − 1.45·21-s − 0.494·22-s − 8.96·23-s + 1.10·24-s + 6.40·25-s − 0.832·26-s + 5.44·27-s − 1.75·28-s + ⋯
L(s)  = 1  + 0.120·2-s − 0.942·3-s − 0.985·4-s − 1.51·5-s − 0.113·6-s + 0.336·7-s − 0.238·8-s − 0.111·9-s − 0.181·10-s − 0.877·11-s + 0.928·12-s − 1.35·13-s + 0.0404·14-s + 1.42·15-s + 0.956·16-s − 0.242·17-s − 0.0134·18-s + 1.66·19-s + 1.48·20-s − 0.317·21-s − 0.105·22-s − 1.86·23-s + 0.224·24-s + 1.28·25-s − 0.163·26-s + 1.04·27-s − 0.331·28-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

\(L(1)\) \(\approx\) \(0.3203618578\)
\(L(\frac12)\) \(\approx\) \(0.3203618578\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.169T + 2T^{2} \)
3 \( 1 + 1.63T + 3T^{2} \)
5 \( 1 + 3.37T + 5T^{2} \)
7 \( 1 - 0.890T + 7T^{2} \)
11 \( 1 + 2.91T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
19 \( 1 - 7.26T + 19T^{2} \)
23 \( 1 + 8.96T + 23T^{2} \)
29 \( 1 - 5.06T + 29T^{2} \)
31 \( 1 - 4.35T + 31T^{2} \)
37 \( 1 + 0.601T + 37T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
47 \( 1 + 3.67T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 + 3.54T + 61T^{2} \)
67 \( 1 - 3.39T + 67T^{2} \)
71 \( 1 + 6.22T + 71T^{2} \)
73 \( 1 + 0.844T + 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + 9.55T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−10.36341842014000570491908562492, −9.751700033885912902348383819094, −8.332233567971384699184518023782, −7.978266639597404486437576847095, −7.00233433006422776440282023583, −5.57068214630569752681675472848, −4.95205663885520368007652791198, −4.19605520192016096243327443772, −2.97127220302087397467172509328, −0.46803698963875958447629706692, 0.46803698963875958447629706692, 2.97127220302087397467172509328, 4.19605520192016096243327443772, 4.95205663885520368007652791198, 5.57068214630569752681675472848, 7.00233433006422776440282023583, 7.978266639597404486437576847095, 8.332233567971384699184518023782, 9.751700033885912902348383819094, 10.36341842014000570491908562492

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