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Properties

Label 2-731-731.16-c1-0-4
Degree $2$
Conductor $731$
Sign $0.990 + 0.138i$
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  + (−0.456 − 2.00i)2-s + (−2.15 − 0.491i)3-s + (−1.99 + 0.959i)4-s + (0.261 − 0.208i)5-s + 4.52i·6-s − 2.19i·7-s + (0.269 + 0.337i)8-s + (1.68 + 0.812i)9-s + (−0.537 − 0.428i)10-s + (−2.76 + 5.74i)11-s + (4.75 − 1.08i)12-s + (0.214 + 0.269i)13-s + (−4.38 + 1.00i)14-s + (−0.665 + 0.320i)15-s + (−2.20 + 2.76i)16-s + (−0.776 − 4.04i)17-s + ⋯
L(s)  = 1  + (−0.322 − 1.41i)2-s + (−1.24 − 0.283i)3-s + (−0.995 + 0.479i)4-s + (0.117 − 0.0933i)5-s + 1.84i·6-s − 0.828i·7-s + (0.0951 + 0.119i)8-s + (0.562 + 0.270i)9-s + (−0.169 − 0.135i)10-s + (−0.833 + 1.73i)11-s + (1.37 − 0.313i)12-s + (0.0595 + 0.0746i)13-s + (−1.17 + 0.267i)14-s + (−0.171 + 0.0827i)15-s + (−0.551 + 0.690i)16-s + (−0.188 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.990 + 0.138i$
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: \((2,\ 731,\ (\ :1/2),\ 0.990 + 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.317149 - 0.0221101i\)
\(L(\frac12)\) \(\approx\) \(0.317149 - 0.0221101i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.776 + 4.04i)T \)
43 \( 1 + (3.34 + 5.63i)T \)
good2 \( 1 + (0.456 + 2.00i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (2.15 + 0.491i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (-0.261 + 0.208i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 + 2.19iT - 7T^{2} \)
11 \( 1 + (2.76 - 5.74i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.214 - 0.269i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (-3.02 + 1.45i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (3.29 - 6.85i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (2.27 - 0.518i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (2.10 - 0.481i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + 2.37iT - 37T^{2} \)
41 \( 1 + (-10.8 + 2.48i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (-7.97 + 3.83i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (3.47 - 4.36i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (6.09 - 7.63i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (9.09 + 2.07i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-2.97 + 1.43i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-2.05 - 4.26i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-6.34 + 5.06i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 1.46iT - 79T^{2} \)
83 \( 1 + (2.22 - 9.76i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (2.94 - 12.9i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-5.54 + 11.5i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.71029632294154748215866133925, −9.688411804068947343354507532583, −9.313736301074013087104683665745, −7.44225819118339520184166884419, −7.15961225544815698700157077840, −5.69337261117757017333576264044, −4.83909504747522319329252023822, −3.76553193313355274045237683374, −2.31905178846346438194271136762, −1.14013382594705420034798462236, 0.24305191448976696722794012639, 2.76247468870227379278189157093, 4.50780053488412182630183903990, 5.68389538630994935299157912266, 5.85939864374648894679379848529, 6.49314078943932727608359325261, 7.993829974887253450235840191253, 8.327593579595604195139372582273, 9.354107223809323840109076701437, 10.50614997071920810639268594897

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