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Properties

Label 2-731-17.16-c1-0-1
Degree $2$
Conductor $731$
Sign $-0.100 + 0.994i$
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  − 0.811·2-s + 2.87i·3-s − 1.34·4-s + 2.95i·5-s − 2.33i·6-s − 2.64i·7-s + 2.71·8-s − 5.25·9-s − 2.39i·10-s + 2.82i·11-s − 3.85i·12-s − 5.05·13-s + 2.14i·14-s − 8.48·15-s + 0.483·16-s + (0.416 − 4.10i)17-s + ⋯
L(s)  = 1  − 0.573·2-s + 1.65i·3-s − 0.670·4-s + 1.32i·5-s − 0.951i·6-s − 1.00i·7-s + 0.958·8-s − 1.75·9-s − 0.757i·10-s + 0.852i·11-s − 1.11i·12-s − 1.40·13-s + 0.573i·14-s − 2.19·15-s + 0.120·16-s + (0.100 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.100 + 0.994i$
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: \((2,\ 731,\ (\ :1/2),\ -0.100 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146952 - 0.162615i\)
\(L(\frac12)\) \(\approx\) \(0.146952 - 0.162615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.416 + 4.10i)T \)
43 \( 1 + T \)
good2 \( 1 + 0.811T + 2T^{2} \)
3 \( 1 - 2.87iT - 3T^{2} \)
5 \( 1 - 2.95iT - 5T^{2} \)
7 \( 1 + 2.64iT - 7T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + 5.05T + 13T^{2} \)
19 \( 1 + 4.73T + 19T^{2} \)
23 \( 1 - 1.01iT - 23T^{2} \)
29 \( 1 - 7.27iT - 29T^{2} \)
31 \( 1 + 6.31iT - 31T^{2} \)
37 \( 1 - 3.34iT - 37T^{2} \)
41 \( 1 + 8.08iT - 41T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 2.52T + 53T^{2} \)
59 \( 1 - 5.42T + 59T^{2} \)
61 \( 1 + 9.03iT - 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 15.0iT - 71T^{2} \)
73 \( 1 - 8.01iT - 73T^{2} \)
79 \( 1 + 7.91iT - 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 3.25T + 89T^{2} \)
97 \( 1 - 6.36iT - 97T^{2} \)
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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

−10.57482868860709262163019081633, −10.09333993023654662542493441100, −9.666902653617545520275249760471, −8.789555346546466540466204890035, −7.46269520632311929487974188221, −7.02234009873356714214802438964, −5.32581309317680130657043295905, −4.48155709309379811034425816678, −3.84778305065157507327704849225, −2.60686503888356191922108853806, 0.14567074855008644622476450096, 1.36576362138889177282969569697, 2.46330733716128322723054292831, 4.39208771777108453232472568269, 5.44482575851236210091182378058, 6.18304977733648570246286766613, 7.46213822115408863577459421228, 8.302107430355304332473925898930, 8.603567174798122186867646815781, 9.347311874720162499109966587392

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