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Properties

Label 2-731-17.16-c1-0-7
Degree $2$
Conductor $731$
Sign $0.993 - 0.110i$
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  − 2.33·2-s − 3.08i·3-s + 3.46·4-s + 3.53i·5-s + 7.20i·6-s − 0.740i·7-s − 3.42·8-s − 6.49·9-s − 8.26i·10-s − 0.0284i·11-s − 10.6i·12-s − 6.23·13-s + 1.73i·14-s + 10.8·15-s + 1.08·16-s + (4.09 − 0.453i)17-s + ⋯
L(s)  = 1  − 1.65·2-s − 1.77i·3-s + 1.73·4-s + 1.58i·5-s + 2.94i·6-s − 0.279i·7-s − 1.21·8-s − 2.16·9-s − 2.61i·10-s − 0.00857i·11-s − 3.08i·12-s − 1.72·13-s + 0.462i·14-s + 2.81·15-s + 0.270·16-s + (0.993 − 0.110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.993 - 0.110i$
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: \((2,\ 731,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.511321 + 0.0282091i\)
\(L(\frac12)\) \(\approx\) \(0.511321 + 0.0282091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-4.09 + 0.453i)T \)
43 \( 1 - T \)
good2 \( 1 + 2.33T + 2T^{2} \)
3 \( 1 + 3.08iT - 3T^{2} \)
5 \( 1 - 3.53iT - 5T^{2} \)
7 \( 1 + 0.740iT - 7T^{2} \)
11 \( 1 + 0.0284iT - 11T^{2} \)
13 \( 1 + 6.23T + 13T^{2} \)
19 \( 1 - 4.71T + 19T^{2} \)
23 \( 1 - 0.160iT - 23T^{2} \)
29 \( 1 - 2.33iT - 29T^{2} \)
31 \( 1 - 5.83iT - 31T^{2} \)
37 \( 1 + 1.79iT - 37T^{2} \)
41 \( 1 - 11.7iT - 41T^{2} \)
47 \( 1 - 6.13T + 47T^{2} \)
53 \( 1 - 4.23T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 4.99T + 67T^{2} \)
71 \( 1 - 7.82iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 + 3.94iT - 79T^{2} \)
83 \( 1 - 8.01T + 83T^{2} \)
89 \( 1 + 5.50T + 89T^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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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

−10.21685573094776919803190661617, −9.675550603323332197031518736672, −8.400620398287493241723181136526, −7.53817634240312092415966564549, −7.20010669959666709097938191288, −6.78792945525400283587578661793, −5.58933321214993850304303476256, −2.99177625549372972698202048120, −2.34552420188977516832164251431, −1.04605252033223700068695239741, 0.55259615366563451647660149446, 2.40996781696320774526689446574, 3.98263597093983053124843982473, 5.05405659901675553217223135656, 5.57770487129450544968346600097, 7.43160929045224766474017817204, 8.207153431178663317523351707123, 9.023434751143947541646761561246, 9.539890742268290507610397904463, 9.898731140598251609076287403680

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