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Properties

Label 2-731-43.16-c1-0-31
Degree $2$
Conductor $731$
Sign $0.0214 + 0.999i$
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  + (−0.335 − 1.47i)2-s + (−0.201 + 0.880i)3-s + (−0.247 + 0.119i)4-s + (−1.49 − 1.87i)5-s + 1.36·6-s + 3.95·7-s + (−1.62 − 2.03i)8-s + (1.96 + 0.947i)9-s + (−2.25 + 2.82i)10-s + (2.02 + 0.973i)11-s + (−0.0552 − 0.242i)12-s + (1.72 + 2.15i)13-s + (−1.32 − 5.81i)14-s + (1.94 − 0.938i)15-s + (−2.78 + 3.49i)16-s + (−0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.237 − 1.03i)2-s + (−0.116 + 0.508i)3-s + (−0.123 + 0.0596i)4-s + (−0.667 − 0.837i)5-s + 0.556·6-s + 1.49·7-s + (−0.573 − 0.719i)8-s + (0.655 + 0.315i)9-s + (−0.712 + 0.892i)10-s + (0.609 + 0.293i)11-s + (−0.0159 − 0.0699i)12-s + (0.477 + 0.598i)13-s + (−0.354 − 1.55i)14-s + (0.503 − 0.242i)15-s + (−0.697 + 0.874i)16-s + (−0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0214 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0214 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.0214 + 0.999i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.0214 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13188 - 1.10784i\)
\(L(\frac12)\) \(\approx\) \(1.13188 - 1.10784i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.0774 + 6.55i)T \)
good2 \( 1 + (0.335 + 1.47i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (0.201 - 0.880i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (1.49 + 1.87i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 - 3.95T + 7T^{2} \)
11 \( 1 + (-2.02 - 0.973i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.72 - 2.15i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (-3.36 + 1.61i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (0.868 + 0.418i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-0.474 - 2.07i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (0.600 + 2.63i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 - 6.44T + 37T^{2} \)
41 \( 1 + (1.92 + 8.42i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (7.42 - 3.57i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-1.35 + 1.70i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-5.89 + 7.39i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-0.373 + 1.63i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (3.62 - 1.74i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-2.80 + 1.34i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-2.74 - 3.43i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + (0.141 - 0.620i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.739 - 3.24i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (5.47 + 2.63i)T + (60.4 + 75.8i)T^{2} \)
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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

−10.33695886241157545747021580255, −9.425549636542415578680825826047, −8.720785016107186111539485430379, −7.81123907058617367470885322369, −6.78937064277571513718635403457, −5.28266626256363759039706443394, −4.39874701875199119012129131427, −3.82380484074635122721091349717, −2.01714289207304108061096787196, −1.12710366730769575813282210655, 1.40250797710949489108450459260, 3.03241317063608772552834317097, 4.31490468833669825082079759745, 5.53996300040265329599973403329, 6.47152821988015839160294770347, 7.20226361456363685566120023154, 7.889948838437629525379976721045, 8.322113748476668811207701522721, 9.568130360853766840427125395653, 10.80372074936174284254927371039

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