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Properties

Label 2-731-731.16-c1-0-19
Degree $2$
Conductor $731$
Sign $-0.664 + 0.747i$
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  + (−0.521 − 2.28i)2-s + (−1.00 − 0.230i)3-s + (−3.14 + 1.51i)4-s + (−2.97 + 2.37i)5-s + 2.42i·6-s + 3.68i·7-s + (2.17 + 2.72i)8-s + (−1.73 − 0.836i)9-s + (6.97 + 5.56i)10-s + (1.17 − 2.44i)11-s + (3.52 − 0.803i)12-s + (0.443 + 0.555i)13-s + (8.42 − 1.92i)14-s + (3.55 − 1.71i)15-s + (0.742 − 0.931i)16-s + (0.494 − 4.09i)17-s + ⋯
L(s)  = 1  + (−0.368 − 1.61i)2-s + (−0.582 − 0.133i)3-s + (−1.57 + 0.756i)4-s + (−1.33 + 1.06i)5-s + 0.990i·6-s + 1.39i·7-s + (0.768 + 0.963i)8-s + (−0.578 − 0.278i)9-s + (2.20 + 1.75i)10-s + (0.354 − 0.736i)11-s + (1.01 − 0.232i)12-s + (0.122 + 0.154i)13-s + (2.25 − 0.513i)14-s + (0.917 − 0.441i)15-s + (0.185 − 0.232i)16-s + (0.119 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.664 + 0.747i$
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: \((2,\ 731,\ (\ :1/2),\ -0.664 + 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185232 - 0.412303i\)
\(L(\frac12)\) \(\approx\) \(0.185232 - 0.412303i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.494 + 4.09i)T \)
43 \( 1 + (-6.34 + 1.67i)T \)
good2 \( 1 + (0.521 + 2.28i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (1.00 + 0.230i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (2.97 - 2.37i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 - 3.68iT - 7T^{2} \)
11 \( 1 + (-1.17 + 2.44i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.443 - 0.555i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (1.54 - 0.743i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.363 + 0.755i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-2.76 + 0.631i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (9.01 - 2.05i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + 6.28iT - 37T^{2} \)
41 \( 1 + (-8.33 + 1.90i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (-5.42 + 2.61i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (6.83 - 8.56i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-4.71 + 5.91i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-3.01 - 0.688i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-1.39 + 0.670i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-2.35 - 4.88i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.05 + 1.63i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 4.91iT - 79T^{2} \)
83 \( 1 + (-2.92 + 12.8i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (2.01 - 8.84i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-2.40 + 4.99i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.56073649217601123899320272936, −9.128529573480443892307719090834, −8.843790752470230953433189165519, −7.68086094148590233044429612152, −6.50877833512537705374161860256, −5.51847467323354261361867699610, −4.03887525301273133464711848849, −3.16165473422173449320948834166, −2.44261569957301689973872464944, −0.43657864535548292851020503556, 0.821503798220472780154156399085, 3.94337495347370924191355336563, 4.52902986640752257029756993892, 5.39801402058693520884530343538, 6.46750386317159558875904188562, 7.38231687196476806909268149517, 7.929645040006169663313795546889, 8.601529564016093913586562300520, 9.533432508399220531101925679497, 10.68537052743145109105874833236

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