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Properties

Label 2-731-17.16-c1-0-41
Degree $2$
Conductor $731$
Sign $-0.889 + 0.456i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.35·2-s − 1.62i·3-s + 3.53·4-s − 0.259i·5-s + 3.81i·6-s − 1.04i·7-s − 3.62·8-s + 0.370·9-s + 0.609i·10-s − 3.29i·11-s − 5.74i·12-s − 0.845·13-s + 2.45i·14-s − 0.420·15-s + 1.45·16-s + (−3.66 + 1.88i)17-s + ⋯
L(s)  = 1  − 1.66·2-s − 0.936i·3-s + 1.76·4-s − 0.115i·5-s + 1.55i·6-s − 0.393i·7-s − 1.28·8-s + 0.123·9-s + 0.192i·10-s − 0.992i·11-s − 1.65i·12-s − 0.234·13-s + 0.654i·14-s − 0.108·15-s + 0.362·16-s + (−0.889 + 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.889 + 0.456i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (560, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.889 + 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124953 - 0.517684i\)
\(L(\frac12)\) \(\approx\) \(0.124953 - 0.517684i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.66 - 1.88i)T \)
43 \( 1 - T \)
good2 \( 1 + 2.35T + 2T^{2} \)
3 \( 1 + 1.62iT - 3T^{2} \)
5 \( 1 + 0.259iT - 5T^{2} \)
7 \( 1 + 1.04iT - 7T^{2} \)
11 \( 1 + 3.29iT - 11T^{2} \)
13 \( 1 + 0.845T + 13T^{2} \)
19 \( 1 - 2.70T + 19T^{2} \)
23 \( 1 + 3.45iT - 23T^{2} \)
29 \( 1 + 6.17iT - 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 - 7.67iT - 37T^{2} \)
41 \( 1 - 3.87iT - 41T^{2} \)
47 \( 1 + 3.53T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + 1.58T + 59T^{2} \)
61 \( 1 + 4.40iT - 61T^{2} \)
67 \( 1 + 9.54T + 67T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + 3.06iT - 73T^{2} \)
79 \( 1 + 0.391iT - 79T^{2} \)
83 \( 1 + 1.93T + 83T^{2} \)
89 \( 1 + 8.91T + 89T^{2} \)
97 \( 1 - 1.22iT - 97T^{2} \)
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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

−9.913108517747582982210905849182, −9.076290571874441859131191904585, −8.181756106095100851277342228309, −7.74404802123517104866228092818, −6.73103722118204441384240026037, −6.22583808694428656002843437480, −4.50310275153154578880241746965, −2.78940930065575605282209033737, −1.56137411813804967185042831173, −0.49841252444025977713456655285, 1.59192081137552467630835805363, 2.89493898644271803658587150916, 4.38803268723795618038583264410, 5.35977406863603520044852892201, 6.97406728130243424831618328633, 7.25452777480787888064535050712, 8.538174740976510484480939615370, 9.287203881654179209102244357058, 9.641302322249447332268018272434, 10.57152545254787680210182880458

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