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Properties

Label 2-728-8.5-c1-0-6
Degree $2$
Conductor $728$
Sign $-0.406 + 0.913i$
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  + (0.529 + 1.31i)2-s + 2.25i·3-s + (−1.43 + 1.38i)4-s + 1.90i·5-s + (−2.95 + 1.19i)6-s − 7-s + (−2.58 − 1.14i)8-s − 2.08·9-s + (−2.50 + 1.01i)10-s − 4.24i·11-s + (−3.13 − 3.24i)12-s + i·13-s + (−0.529 − 1.31i)14-s − 4.30·15-s + (0.137 − 3.99i)16-s − 6.18·17-s + ⋯
L(s)  = 1  + (0.374 + 0.927i)2-s + 1.30i·3-s + (−0.719 + 0.694i)4-s + 0.853i·5-s + (−1.20 + 0.487i)6-s − 0.377·7-s + (−0.913 − 0.406i)8-s − 0.693·9-s + (−0.791 + 0.319i)10-s − 1.28i·11-s + (−0.904 − 0.935i)12-s + 0.277i·13-s + (−0.141 − 0.350i)14-s − 1.11·15-s + (0.0343 − 0.999i)16-s − 1.50·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.406 + 0.913i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.406 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.619946 - 0.954198i\)
\(L(\frac12)\) \(\approx\) \(0.619946 - 0.954198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.529 - 1.31i)T \)
7 \( 1 + T \)
13 \( 1 - iT \)
good3 \( 1 - 2.25iT - 3T^{2} \)
5 \( 1 - 1.90iT - 5T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
17 \( 1 + 6.18T + 17T^{2} \)
19 \( 1 - 6.45iT - 19T^{2} \)
23 \( 1 - 1.86T + 23T^{2} \)
29 \( 1 - 2.91iT - 29T^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 - 3.46iT - 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 7.62iT - 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 - 8.53iT - 53T^{2} \)
59 \( 1 + 1.16iT - 59T^{2} \)
61 \( 1 - 9.61iT - 61T^{2} \)
67 \( 1 - 4.60iT - 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 + 16.0iT - 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 - 2.39T + 97T^{2} \)
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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

−10.81385793786301139968441044005, −10.11583537159304342476027576354, −9.090115373985843139326522644831, −8.590156048019357816160182579824, −7.34991184755883372252272130154, −6.42252763919606350034550735809, −5.72106251518815059796386684399, −4.59191113522592438859576922590, −3.72419579010678769146874464369, −2.99264885399693898029626789726, 0.51490562610752252741115060382, 1.81096573401046518242646885665, 2.68448411673600866210627714263, 4.36131621489019014578694085108, 4.99521426831509997702065424895, 6.34403907412269384948316640507, 7.03846317922090147572136039244, 8.179634729692277107873926935748, 9.133160390614296302300025799090, 9.692556470606672454848163390940

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