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Properties

Label 2-728-104.101-c1-0-23
Degree $2$
Conductor $728$
Sign $0.794 - 0.607i$
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 + (−0.836 + 0.482i)3-s + (−1.43 − 1.38i)4-s − 3.93·5-s + (−0.189 − 1.35i)6-s + (−0.866 − 0.5i)7-s + (2.58 − 1.14i)8-s + (−1.03 + 1.79i)9-s + (2.08 − 5.15i)10-s + (−0.864 − 1.49i)11-s + (1.87 + 0.467i)12-s + (−3.49 + 0.882i)13-s + (1.11 − 0.870i)14-s + (3.29 − 1.89i)15-s + (0.136 + 3.99i)16-s + (1.48 − 2.56i)17-s + ⋯
L(s)  = 1  + (−0.374 + 0.927i)2-s + (−0.482 + 0.278i)3-s + (−0.719 − 0.694i)4-s − 1.75·5-s + (−0.0774 − 0.552i)6-s + (−0.327 − 0.188i)7-s + (0.913 − 0.406i)8-s + (−0.344 + 0.596i)9-s + (0.659 − 1.63i)10-s + (−0.260 − 0.451i)11-s + (0.540 + 0.135i)12-s + (−0.969 + 0.244i)13-s + (0.297 − 0.232i)14-s + (0.849 − 0.490i)15-s + (0.0342 + 0.999i)16-s + (0.359 − 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.794 - 0.607i$
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: \((2,\ 728,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.372401 + 0.126057i\)
\(L(\frac12)\) \(\approx\) \(0.372401 + 0.126057i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (3.49 - 0.882i)T \)
good3 \( 1 + (0.836 - 0.482i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.93T + 5T^{2} \)
11 \( 1 + (0.864 + 1.49i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.48 + 2.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.479 + 0.831i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.176 - 0.305i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0559 + 0.0322i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.86iT - 31T^{2} \)
37 \( 1 + (-3.57 - 6.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.88 + 2.24i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.27 - 5.35i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.29iT - 47T^{2} \)
53 \( 1 + 8.51iT - 53T^{2} \)
59 \( 1 + (0.0552 - 0.0957i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.85 + 5.11i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.49 - 6.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.5 + 6.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 15.5iT - 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 + (-2.95 + 1.70i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.13 + 1.80i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.51537598790076683514027815280, −9.516565590382181994336233690434, −8.530463507894280665601118802523, −7.70443942238173226713645803891, −7.29798712418913494370027214234, −6.14386133318694160943712425395, −4.95784210598627030476699783375, −4.45598545463902458988187783656, −3.11130500424576463461339405523, −0.45513703818568126683493543226, 0.66046302928153947233567083344, 2.63877378767878765330840361503, 3.69760799561200980345573909502, 4.46906954568067972285046655399, 5.74270559535789787636447645744, 7.25708877943843916983275711799, 7.67164421450413369533407562721, 8.653700791787710674949947103272, 9.506622839875506561426866325436, 10.52402479668032932870625442821

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