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Properties

Label 2-731-731.16-c1-0-21
Degree $2$
Conductor $731$
Sign $0.966 - 0.256i$
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  + (−0.343 − 1.50i)2-s + (2.15 + 0.492i)3-s + (−0.349 + 0.168i)4-s + (−2.57 + 2.05i)5-s − 3.41i·6-s + 1.34i·7-s + (−1.55 − 1.94i)8-s + (1.70 + 0.822i)9-s + (3.97 + 3.17i)10-s + (−1.74 + 3.62i)11-s + (−0.837 + 0.191i)12-s + (3.19 + 4.01i)13-s + (2.02 − 0.462i)14-s + (−6.56 + 3.15i)15-s + (−2.88 + 3.61i)16-s + (3.77 + 1.66i)17-s + ⋯
L(s)  = 1  + (−0.243 − 1.06i)2-s + (1.24 + 0.284i)3-s + (−0.174 + 0.0842i)4-s + (−1.15 + 0.917i)5-s − 1.39i·6-s + 0.507i·7-s + (−0.549 − 0.688i)8-s + (0.569 + 0.274i)9-s + (1.25 + 1.00i)10-s + (−0.525 + 1.09i)11-s + (−0.241 + 0.0552i)12-s + (0.887 + 1.11i)13-s + (0.541 − 0.123i)14-s + (−1.69 + 0.815i)15-s + (−0.721 + 0.904i)16-s + (0.914 + 0.403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.966 - 0.256i$
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: \((2,\ 731,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64125 + 0.213641i\)
\(L(\frac12)\) \(\approx\) \(1.64125 + 0.213641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.77 - 1.66i)T \)
43 \( 1 + (0.211 - 6.55i)T \)
good2 \( 1 + (0.343 + 1.50i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-2.15 - 0.492i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (2.57 - 2.05i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 - 1.34iT - 7T^{2} \)
11 \( 1 + (1.74 - 3.62i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-3.19 - 4.01i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (-6.58 + 3.17i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (2.59 - 5.37i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-2.98 + 0.680i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (0.750 - 0.171i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + 4.93iT - 37T^{2} \)
41 \( 1 + (-6.19 + 1.41i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (6.77 - 3.26i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (4.22 - 5.29i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (2.14 - 2.69i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (11.2 + 2.57i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (2.13 - 1.02i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (5.80 + 12.0i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-6.69 + 5.34i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 6.88iT - 79T^{2} \)
83 \( 1 + (-3.58 + 15.7i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (2.20 - 9.65i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (0.00894 - 0.0185i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.41848870070752309085582999904, −9.504589596324182302178930325129, −9.101516862356069906355154606139, −7.80481280253353184350043420484, −7.38794424288541341640704977683, −6.11063503245161876853780137041, −4.34966005672037044818114403834, −3.41040108501651677728901196298, −2.92809644391507616122226008303, −1.76819439004093660736554302180, 0.830625007391342814334602598157, 3.00386718220380499443987495321, 3.60053525235667644898347974619, 5.11748102630057063659592553775, 5.99648813949483123889155326422, 7.41385166502703342103175761375, 7.904032301211552970224358520505, 8.305850363251661184035964035179, 8.851755674540146733283248250499, 10.11726855283704541406627586749

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