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Properties

Label 2-731-731.16-c1-0-47
Degree $2$
Conductor $731$
Sign $-0.967 - 0.254i$
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  + (−0.343 − 1.50i)2-s + (−2.15 − 0.492i)3-s + (−0.349 + 0.168i)4-s + (2.57 − 2.05i)5-s + 3.41i·6-s − 1.34i·7-s + (−1.55 − 1.94i)8-s + (1.70 + 0.822i)9-s + (−3.97 − 3.17i)10-s + (1.74 − 3.62i)11-s + (0.837 − 0.191i)12-s + (3.19 + 4.01i)13-s + (−2.02 + 0.462i)14-s + (−6.56 + 3.15i)15-s + (−2.88 + 3.61i)16-s + (−2.46 − 3.30i)17-s + ⋯
L(s)  = 1  + (−0.243 − 1.06i)2-s + (−1.24 − 0.284i)3-s + (−0.174 + 0.0842i)4-s + (1.15 − 0.917i)5-s + 1.39i·6-s − 0.507i·7-s + (−0.549 − 0.688i)8-s + (0.569 + 0.274i)9-s + (−1.25 − 1.00i)10-s + (0.525 − 1.09i)11-s + (0.241 − 0.0552i)12-s + (0.887 + 1.11i)13-s + (−0.541 + 0.123i)14-s + (−1.69 + 0.815i)15-s + (−0.721 + 0.904i)16-s + (−0.596 − 0.802i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.967 - 0.254i$
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: \((2,\ 731,\ (\ :1/2),\ -0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145740 + 1.12858i\)
\(L(\frac12)\) \(\approx\) \(0.145740 + 1.12858i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.46 + 3.30i)T \)
43 \( 1 + (0.211 - 6.55i)T \)
good2 \( 1 + (0.343 + 1.50i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (2.15 + 0.492i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (-2.57 + 2.05i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 + 1.34iT - 7T^{2} \)
11 \( 1 + (-1.74 + 3.62i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-3.19 - 4.01i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (-6.58 + 3.17i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-2.59 + 5.37i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (2.98 - 0.680i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-0.750 + 0.171i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 - 4.93iT - 37T^{2} \)
41 \( 1 + (6.19 - 1.41i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (6.77 - 3.26i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (4.22 - 5.29i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (2.14 - 2.69i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-11.2 - 2.57i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (2.13 - 1.02i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-5.80 - 12.0i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (6.69 - 5.34i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 6.88iT - 79T^{2} \)
83 \( 1 + (-3.58 + 15.7i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (2.20 - 9.65i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-0.00894 + 0.0185i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.07288257426256602219357854627, −9.232000833110036681269027366791, −8.758249738131459038655282115825, −6.84320148931449779525758184392, −6.36798117147149729690346971044, −5.46675633630307099779005466411, −4.47970267482222564996307423172, −2.99843004490956789114378566118, −1.41667231850476763704302615409, −0.824207780292007069209531183400, 1.91914039541335931679834658926, 3.41088333100714985244741310104, 5.35635897579646741539393576241, 5.56250750412868845979528720501, 6.40439574205714978297465841075, 7.00908262110902084043413345003, 8.061703870816861614709131167595, 9.242314469585822649990570922579, 10.00001586467490978891138871764, 10.78343324312560917111603426236

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