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Properties

Label 2-731-17.16-c1-0-47
Degree $2$
Conductor $731$
Sign $0.635 + 0.772i$
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  + 0.887·2-s + 1.66i·3-s − 1.21·4-s + 0.927i·5-s + 1.47i·6-s − 4.90i·7-s − 2.85·8-s + 0.241·9-s + 0.823i·10-s − 6.10i·11-s − 2.01i·12-s + 2.16·13-s − 4.35i·14-s − 1.54·15-s − 0.108·16-s + (−2.62 − 3.18i)17-s + ⋯
L(s)  = 1  + 0.627·2-s + 0.958i·3-s − 0.605·4-s + 0.414i·5-s + 0.602i·6-s − 1.85i·7-s − 1.00·8-s + 0.0803·9-s + 0.260i·10-s − 1.84i·11-s − 0.581i·12-s + 0.599·13-s − 1.16i·14-s − 0.397·15-s − 0.0270·16-s + (−0.635 − 0.772i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.635 + 0.772i$
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: \((2,\ 731,\ (\ :1/2),\ 0.635 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32774 - 0.626739i\)
\(L(\frac12)\) \(\approx\) \(1.32774 - 0.626739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.62 + 3.18i)T \)
43 \( 1 + T \)
good2 \( 1 - 0.887T + 2T^{2} \)
3 \( 1 - 1.66iT - 3T^{2} \)
5 \( 1 - 0.927iT - 5T^{2} \)
7 \( 1 + 4.90iT - 7T^{2} \)
11 \( 1 + 6.10iT - 11T^{2} \)
13 \( 1 - 2.16T + 13T^{2} \)
19 \( 1 + 2.62T + 19T^{2} \)
23 \( 1 - 7.02iT - 23T^{2} \)
29 \( 1 + 8.89iT - 29T^{2} \)
31 \( 1 - 3.66iT - 31T^{2} \)
37 \( 1 + 2.53iT - 37T^{2} \)
41 \( 1 + 11.4iT - 41T^{2} \)
47 \( 1 - 3.63T + 47T^{2} \)
53 \( 1 - 4.63T + 53T^{2} \)
59 \( 1 + 6.50T + 59T^{2} \)
61 \( 1 + 1.99iT - 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 - 5.47iT - 71T^{2} \)
73 \( 1 + 3.49iT - 73T^{2} \)
79 \( 1 + 4.17iT - 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 4.80iT - 97T^{2} \)
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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

−10.58955142614728100465941556163, −9.453155423602556434828885782712, −8.765013542483325722220895845564, −7.63405174923273031565329381121, −6.59648424527894402933774672442, −5.58564262746129735042900636838, −4.52270267604084541854277387066, −3.79727946844292232619213100469, −3.30363272113672829763184129285, −0.65609484414522463332021280993, 1.70576897720179210475333931502, 2.69195144231423119080533015186, 4.38601387388286185587986341710, 4.96447947192312906059173151113, 6.14302598680525139157336396137, 6.69139913380034411112269391520, 8.092127595522792909032401625116, 8.766522991072370474010029800488, 9.357697912241811468788794412545, 10.50979795225109691748238669830

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