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Properties

Label 2-728-104.101-c1-0-27
Degree $2$
Conductor $728$
Sign $0.923 + 0.383i$
Analytic cond. $5.81310$
Root an. cond. $2.41103$
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  + (0.687 − 1.23i)2-s + (−1.40 + 0.809i)3-s + (−1.05 − 1.69i)4-s − 2.10·5-s + (0.0358 + 2.28i)6-s + (−0.866 − 0.5i)7-s + (−2.82 + 0.132i)8-s + (−0.190 + 0.330i)9-s + (−1.44 + 2.59i)10-s + (2.17 + 3.76i)11-s + (2.85 + 1.52i)12-s + (3.28 + 1.49i)13-s + (−1.21 + 0.726i)14-s + (2.94 − 1.70i)15-s + (−1.77 + 3.58i)16-s + (2.81 − 4.87i)17-s + ⋯
L(s)  = 1  + (0.486 − 0.873i)2-s + (−0.809 + 0.467i)3-s + (−0.526 − 0.849i)4-s − 0.940·5-s + (0.0146 + 0.934i)6-s + (−0.327 − 0.188i)7-s + (−0.998 + 0.0469i)8-s + (−0.0636 + 0.110i)9-s + (−0.457 + 0.821i)10-s + (0.655 + 1.13i)11-s + (0.823 + 0.441i)12-s + (0.909 + 0.414i)13-s + (−0.324 + 0.194i)14-s + (0.760 − 0.439i)15-s + (−0.444 + 0.895i)16-s + (0.682 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06491 - 0.212346i\)
\(L(\frac12)\) \(\approx\) \(1.06491 - 0.212346i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.687 + 1.23i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-3.28 - 1.49i)T \)
good3 \( 1 + (1.40 - 0.809i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.10T + 5T^{2} \)
11 \( 1 + (-2.17 - 3.76i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.81 + 4.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.13 + 7.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.04 - 5.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.05 + 0.609i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.58iT - 31T^{2} \)
37 \( 1 + (1.96 + 3.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.53 + 2.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.82 - 4.52i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.60iT - 47T^{2} \)
53 \( 1 - 7.82iT - 53T^{2} \)
59 \( 1 + (3.60 - 6.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.27 - 1.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.990 - 1.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.74 - 3.89i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.30iT - 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 + (3.71 - 2.14i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.53 + 2.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.73648082825423416783643421447, −9.499530262607997576727409947995, −9.190371333166727499430666269064, −7.58350925118248985119230665949, −6.73557253543694378350327536474, −5.47614549781422740246415183479, −4.73384599514280157587526083629, −3.94794527682078830460641266474, −2.87313756505631227165150667034, −0.996533975007250013832133228443, 0.75371344320994192464180976210, 3.48136353320549791070632837452, 3.79947716929888785185269443363, 5.46565026989535750450445851466, 6.05367815905199268342621579702, 6.62882058390445998477612007350, 7.927026571353798390885645364917, 8.267619581235223800324750961037, 9.340757849669520238298144282056, 10.71809018853225300408064578181

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