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Properties

Label 2-731-731.16-c1-0-50
Degree $2$
Conductor $731$
Sign $-0.414 - 0.909i$
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  + (−0.608 − 2.66i)2-s + (0.149 + 0.0342i)3-s + (−4.94 + 2.38i)4-s + (1.75 − 1.40i)5-s − 0.420i·6-s + 1.31i·7-s + (5.95 + 7.46i)8-s + (−2.68 − 1.29i)9-s + (−4.81 − 3.83i)10-s + (2.12 − 4.40i)11-s + (−0.823 + 0.187i)12-s + (−2.22 − 2.79i)13-s + (3.52 − 0.803i)14-s + (0.311 − 0.150i)15-s + (9.44 − 11.8i)16-s + (−3.58 − 2.04i)17-s + ⋯
L(s)  = 1  + (−0.430 − 1.88i)2-s + (0.0865 + 0.0197i)3-s + (−2.47 + 1.19i)4-s + (0.786 − 0.626i)5-s − 0.171i·6-s + 0.498i·7-s + (2.10 + 2.63i)8-s + (−0.893 − 0.430i)9-s + (−1.52 − 1.21i)10-s + (0.639 − 1.32i)11-s + (−0.237 + 0.0542i)12-s + (−0.618 − 0.775i)13-s + (0.940 − 0.214i)14-s + (0.0804 − 0.0387i)15-s + (2.36 − 2.96i)16-s + (−0.868 − 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.414 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.414 - 0.909i$
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: \((2,\ 731,\ (\ :1/2),\ -0.414 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396960 + 0.617288i\)
\(L(\frac12)\) \(\approx\) \(0.396960 + 0.617288i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.58 + 2.04i)T \)
43 \( 1 + (6.36 - 1.55i)T \)
good2 \( 1 + (0.608 + 2.66i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-0.149 - 0.0342i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (-1.75 + 1.40i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 - 1.31iT - 7T^{2} \)
11 \( 1 + (-2.12 + 4.40i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.22 + 2.79i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (-5.01 + 2.41i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (3.11 - 6.46i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (8.59 - 1.96i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (0.464 - 0.106i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + 8.56iT - 37T^{2} \)
41 \( 1 + (2.28 - 0.522i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (-2.59 + 1.25i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.32 - 1.66i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.20 - 1.51i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (3.87 + 0.883i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-2.25 + 1.08i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (4.57 + 9.49i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-9.89 + 7.88i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 5.49iT - 79T^{2} \)
83 \( 1 + (3.24 - 14.2i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-0.512 + 2.24i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (4.23 - 8.78i)T + (-60.4 - 75.8i)T^{2} \)
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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

−9.562835078390538598204959721733, −9.251521599774076517910203458989, −8.749305357413325457910267039261, −7.67167316780810275834597956465, −5.69477527559039153852951604562, −5.22056515555070190805828618416, −3.66087608162964464483389743310, −2.94201285265907860807982805360, −1.79328807012925096771280729570, −0.43005727409062919868246039571, 1.97812982473820604953241901748, 4.08502352949436294848581624457, 4.96948620210337713364189352431, 6.00250277095505553224798062161, 6.70095929974229570422023025839, 7.30670484895934353321961631097, 8.209277507036814459845793615387, 9.137660217335688893768309486842, 9.850184498971865723018805933458, 10.37952033224885596401115971348

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