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Properties

Label 2-731-17.16-c1-0-50
Degree $2$
Conductor $731$
Sign $0.731 + 0.681i$
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  + 2.57·2-s − 2.25i·3-s + 4.62·4-s + 3.77i·5-s − 5.81i·6-s − 3.81i·7-s + 6.75·8-s − 2.10·9-s + 9.72i·10-s + 0.653i·11-s − 10.4i·12-s − 0.649·13-s − 9.82i·14-s + 8.53·15-s + 8.13·16-s + (−3.01 − 2.81i)17-s + ⋯
L(s)  = 1  + 1.81·2-s − 1.30i·3-s + 2.31·4-s + 1.68i·5-s − 2.37i·6-s − 1.44i·7-s + 2.38·8-s − 0.700·9-s + 3.07i·10-s + 0.196i·11-s − 3.01i·12-s − 0.180·13-s − 2.62i·14-s + 2.20·15-s + 2.03·16-s + (−0.731 − 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.731 + 0.681i$
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: \((2,\ 731,\ (\ :1/2),\ 0.731 + 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.08843 - 1.60918i\)
\(L(\frac12)\) \(\approx\) \(4.08843 - 1.60918i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.01 + 2.81i)T \)
43 \( 1 + T \)
good2 \( 1 - 2.57T + 2T^{2} \)
3 \( 1 + 2.25iT - 3T^{2} \)
5 \( 1 - 3.77iT - 5T^{2} \)
7 \( 1 + 3.81iT - 7T^{2} \)
11 \( 1 - 0.653iT - 11T^{2} \)
13 \( 1 + 0.649T + 13T^{2} \)
19 \( 1 - 4.42T + 19T^{2} \)
23 \( 1 - 5.66iT - 23T^{2} \)
29 \( 1 - 7.17iT - 29T^{2} \)
31 \( 1 + 3.17iT - 31T^{2} \)
37 \( 1 - 7.38iT - 37T^{2} \)
41 \( 1 + 4.29iT - 41T^{2} \)
47 \( 1 + 4.56T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 - 3.01T + 59T^{2} \)
61 \( 1 - 2.54iT - 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 - 5.19iT - 71T^{2} \)
73 \( 1 - 14.0iT - 73T^{2} \)
79 \( 1 + 16.4iT - 79T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 + 0.705T + 89T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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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

−10.76936466176153539665868684503, −9.875707305398113539453060958159, −7.64013112644264355534557900285, −7.21601853473824061024205672545, −6.84084155465743933283940824762, −6.05042756791009132719692600245, −4.77526145050843903820846607462, −3.56764703256439766708379525456, −2.90015317257446852911620255891, −1.68126249260050519767142100640, 2.07143892906267227956312148725, 3.34116540985540747162005239374, 4.43873603747226516655364897453, 4.84326555392359844182194547174, 5.56690634101025209728537778899, 6.27749135458842025245556973377, 8.017792174025551306576697968614, 8.943021044981418823898728516229, 9.562092182172738321342539880264, 10.77779191383243680599369760323

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