Properties

Label 2-731-1.1-c1-0-7
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  − 1.62·2-s − 3.32·3-s + 0.632·4-s + 2.56·5-s + 5.39·6-s − 2.29·7-s + 2.21·8-s + 8.06·9-s − 4.16·10-s + 2.13·11-s − 2.10·12-s − 3.41·13-s + 3.71·14-s − 8.53·15-s − 4.86·16-s − 17-s − 13.0·18-s − 1.31·19-s + 1.62·20-s + 7.62·21-s − 3.46·22-s − 1.30·23-s − 7.37·24-s + 1.58·25-s + 5.53·26-s − 16.8·27-s − 1.45·28-s + ⋯
L(s)  = 1  − 1.14·2-s − 1.92·3-s + 0.316·4-s + 1.14·5-s + 2.20·6-s − 0.866·7-s + 0.784·8-s + 2.68·9-s − 1.31·10-s + 0.643·11-s − 0.607·12-s − 0.946·13-s + 0.994·14-s − 2.20·15-s − 1.21·16-s − 0.242·17-s − 3.08·18-s − 0.302·19-s + 0.363·20-s + 1.66·21-s − 0.738·22-s − 0.272·23-s − 1.50·24-s + 0.317·25-s + 1.08·26-s − 3.24·27-s − 0.274·28-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

\(L(1)\) \(\approx\) \(0.3812744359\)
\(L(\frac12)\) \(\approx\) \(0.3812744359\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.62T + 2T^{2} \)
3 \( 1 + 3.32T + 3T^{2} \)
5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 + 2.29T + 7T^{2} \)
11 \( 1 - 2.13T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
19 \( 1 + 1.31T + 19T^{2} \)
23 \( 1 + 1.30T + 23T^{2} \)
29 \( 1 - 1.44T + 29T^{2} \)
31 \( 1 - 0.817T + 31T^{2} \)
37 \( 1 - 1.28T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
47 \( 1 + 6.26T + 47T^{2} \)
53 \( 1 - 9.09T + 53T^{2} \)
59 \( 1 - 0.749T + 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + 8.53T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 - 3.41T + 79T^{2} \)
83 \( 1 - 1.16T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−10.19266441073723737245466456240, −9.736795930443999058731120570273, −9.170194542575372995616860393431, −7.62133745881329053811393794324, −6.68372956780870073094232106605, −6.19437554522288114688554377355, −5.21127568487309324627574051047, −4.26530657310337211710339307394, −1.99007971383590425988855886613, −0.67585889656602762824869855488, 0.67585889656602762824869855488, 1.99007971383590425988855886613, 4.26530657310337211710339307394, 5.21127568487309324627574051047, 6.19437554522288114688554377355, 6.68372956780870073094232106605, 7.62133745881329053811393794324, 9.170194542575372995616860393431, 9.736795930443999058731120570273, 10.19266441073723737245466456240

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