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Properties

Label 2-731-17.16-c1-0-56
Degree $2$
Conductor $731$
Sign $-0.861 + 0.508i$
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  + 1.49·2-s − 2.58i·3-s + 0.247·4-s + 1.07i·5-s − 3.88i·6-s − 1.49i·7-s − 2.62·8-s − 3.70·9-s + 1.60i·10-s − 3.55i·11-s − 0.641i·12-s − 3.17·13-s − 2.23i·14-s + 2.77·15-s − 4.43·16-s + (3.55 − 2.09i)17-s + ⋯
L(s)  = 1  + 1.06·2-s − 1.49i·3-s + 0.123·4-s + 0.479i·5-s − 1.58i·6-s − 0.564i·7-s − 0.928·8-s − 1.23·9-s + 0.508i·10-s − 1.07i·11-s − 0.185i·12-s − 0.881·13-s − 0.598i·14-s + 0.716·15-s − 1.10·16-s + (0.861 − 0.508i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.861 + 0.508i$
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: \((2,\ 731,\ (\ :1/2),\ -0.861 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.460675 - 1.68569i\)
\(L(\frac12)\) \(\approx\) \(0.460675 - 1.68569i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.55 + 2.09i)T \)
43 \( 1 + T \)
good2 \( 1 - 1.49T + 2T^{2} \)
3 \( 1 + 2.58iT - 3T^{2} \)
5 \( 1 - 1.07iT - 5T^{2} \)
7 \( 1 + 1.49iT - 7T^{2} \)
11 \( 1 + 3.55iT - 11T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
19 \( 1 + 4.85T + 19T^{2} \)
23 \( 1 + 1.35iT - 23T^{2} \)
29 \( 1 - 3.45iT - 29T^{2} \)
31 \( 1 + 8.25iT - 31T^{2} \)
37 \( 1 - 1.27iT - 37T^{2} \)
41 \( 1 - 7.69iT - 41T^{2} \)
47 \( 1 - 1.52T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 0.438T + 59T^{2} \)
61 \( 1 + 1.23iT - 61T^{2} \)
67 \( 1 + 3.47T + 67T^{2} \)
71 \( 1 + 9.97iT - 71T^{2} \)
73 \( 1 + 1.44iT - 73T^{2} \)
79 \( 1 + 9.00iT - 79T^{2} \)
83 \( 1 + 2.25T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 11.2iT - 97T^{2} \)
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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

−10.22371662097118584366069144474, −8.981825427196739746192323725165, −8.030208203230912707729162589115, −7.19428598320638697952527030848, −6.43204933855129472786478936005, −5.71906488311807411828442182903, −4.55529962399673180656425768654, −3.30563902767544451637062787915, −2.41976597032865342202726991444, −0.62687060244005915595428089561, 2.46036710219592705654187663737, 3.68733996294782717175047069524, 4.44576707778463131436846563244, 5.09413689355405345844770611156, 5.72326613249631598677152448390, 7.05966915560450958808264304181, 8.560510945865833770502753581830, 9.097091370986185368603949419329, 9.971195255222000206443580355640, 10.54569230998849994222216275763

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