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Properties

Label 2-731-731.16-c1-0-53
Degree $2$
Conductor $731$
Sign $0.0231 + 0.999i$
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  + (−0.332 − 1.45i)2-s + (2.23 + 0.509i)3-s + (−0.211 + 0.101i)4-s + (3.41 − 2.72i)5-s − 3.41i·6-s + 3.16i·7-s + (−1.64 − 2.06i)8-s + (2.01 + 0.968i)9-s + (−5.10 − 4.06i)10-s + (1.29 − 2.69i)11-s + (−0.523 + 0.119i)12-s + (1.50 + 1.89i)13-s + (4.60 − 1.05i)14-s + (8.99 − 4.33i)15-s + (−2.75 + 3.45i)16-s + (−3.59 + 2.02i)17-s + ⋯
L(s)  = 1  + (−0.235 − 1.03i)2-s + (1.28 + 0.293i)3-s + (−0.105 + 0.0509i)4-s + (1.52 − 1.21i)5-s − 1.39i·6-s + 1.19i·7-s + (−0.581 − 0.729i)8-s + (0.670 + 0.322i)9-s + (−1.61 − 1.28i)10-s + (0.391 − 0.813i)11-s + (−0.151 + 0.0345i)12-s + (0.418 + 0.524i)13-s + (1.23 − 0.281i)14-s + (2.32 − 1.11i)15-s + (−0.688 + 0.862i)16-s + (−0.871 + 0.490i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0231 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0231 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.0231 + 0.999i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.0231 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92498 - 1.88088i\)
\(L(\frac12)\) \(\approx\) \(1.92498 - 1.88088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.59 - 2.02i)T \)
43 \( 1 + (-2.40 - 6.10i)T \)
good2 \( 1 + (0.332 + 1.45i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-2.23 - 0.509i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (-3.41 + 2.72i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 - 3.16iT - 7T^{2} \)
11 \( 1 + (-1.29 + 2.69i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-1.50 - 1.89i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (3.69 - 1.78i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (2.53 - 5.25i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-3.59 + 0.820i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (7.49 - 1.70i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 - 10.7iT - 37T^{2} \)
41 \( 1 + (-1.41 + 0.323i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (-9.11 + 4.38i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.174 + 0.218i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.28 - 1.61i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (13.2 + 3.01i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (1.16 - 0.560i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (4.49 + 9.33i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-9.86 + 7.86i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 2.35iT - 79T^{2} \)
83 \( 1 + (0.329 - 1.44i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-1.62 + 7.13i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-6.95 + 14.4i)T + (-60.4 - 75.8i)T^{2} \)
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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

−9.904771198266031417797091918323, −9.160041770937428167601287505463, −8.937276365651003035142329480915, −8.327402697675050187011459886237, −6.26411292371965363268671872120, −5.84946678957556441136379568246, −4.38878424490571965971002085753, −3.18986141658164847012125856971, −2.12828770152581143927758764072, −1.62417197915513677669645876959, 2.09913151402755219492566613396, 2.66680483241512373818590598568, 4.02705654405663362195959889880, 5.65696012472531028383430386349, 6.63630113957603849064548897955, 7.07678042297393967309510677050, 7.72246073737967669165589512085, 8.924373095137203379911166802971, 9.369344224438026310883223466880, 10.59577522374092310713671339941

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