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Properties

Label 2-731-43.35-c1-0-2
Degree $2$
Conductor $731$
Sign $-0.176 + 0.984i$
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  + (−0.396 + 1.73i)2-s + (−0.0213 − 0.0934i)3-s + (−1.05 − 0.508i)4-s + (−1.30 + 1.63i)5-s + 0.170·6-s − 1.70·7-s + (−0.920 + 1.15i)8-s + (2.69 − 1.29i)9-s + (−2.31 − 2.90i)10-s + (−1.66 + 0.802i)11-s + (−0.0249 + 0.109i)12-s + (−1.34 + 1.68i)13-s + (0.673 − 2.95i)14-s + (0.180 + 0.0868i)15-s + (−3.09 − 3.88i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.280 + 1.22i)2-s + (−0.0123 − 0.0539i)3-s + (−0.527 − 0.254i)4-s + (−0.582 + 0.730i)5-s + 0.0697·6-s − 0.642·7-s + (−0.325 + 0.407i)8-s + (0.898 − 0.432i)9-s + (−0.733 − 0.919i)10-s + (−0.502 + 0.241i)11-s + (−0.00721 + 0.0316i)12-s + (−0.371 + 0.466i)13-s + (0.180 − 0.789i)14-s + (0.0465 + 0.0224i)15-s + (−0.774 − 0.971i)16-s + (0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.176 + 0.984i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.176 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208851 - 0.249693i\)
\(L(\frac12)\) \(\approx\) \(0.208851 - 0.249693i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.943 - 6.48i)T \)
good2 \( 1 + (0.396 - 1.73i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.0213 + 0.0934i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (1.30 - 1.63i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + 1.70T + 7T^{2} \)
11 \( 1 + (1.66 - 0.802i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (1.34 - 1.68i)T + (-2.89 - 12.6i)T^{2} \)
19 \( 1 + (5.46 + 2.63i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (1.05 - 0.509i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-2.35 + 10.3i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (1.69 - 7.41i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + 6.51T + 37T^{2} \)
41 \( 1 + (-2.34 + 10.2i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-1.05 - 0.510i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (3.38 + 4.24i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-5.16 - 6.47i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (0.727 + 3.18i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (4.62 + 2.22i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (1.28 + 0.619i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (1.16 - 1.46i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (-3.78 - 16.5i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-3.19 - 14.0i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-10.0 + 4.83i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.84372538166617625442719849475, −10.00149456783774021258394700925, −9.131488207186341778037768848493, −8.180874808701177395454914773626, −7.28709245419857801677555050722, −6.82601721458588020978708306066, −6.13831292953443848320492794656, −4.83051644198078250358328088306, −3.71658711877980135134656553303, −2.41208247322299613103317356627, 0.17616534340601426056065702457, 1.66117549883108052580899469284, 2.93965180887657467350870448747, 3.96981992231945415839911058689, 4.87522272338516292788408294152, 6.20617508205233637845678716831, 7.31233290203783533961962909040, 8.309875844115762689219826832441, 9.091147299887974657862578259943, 10.19242622659589401126867486355

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