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Properties

Label 2-728-728.51-c0-0-3
Degree $2$
Conductor $728$
Sign $0.993 + 0.110i$
Analytic cond. $0.363319$
Root an. cond. $0.602759$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.939 + 1.62i)3-s + (−0.499 − 0.866i)4-s + (0.173 − 0.300i)5-s + 1.87·6-s + (0.939 − 0.342i)7-s − 0.999·8-s + (−1.26 + 2.19i)9-s + (−0.173 − 0.300i)10-s + (0.939 − 1.62i)12-s − 13-s + (0.173 − 0.984i)14-s + 0.652·15-s + (−0.5 + 0.866i)16-s + (−0.766 − 1.32i)17-s + (1.26 + 2.19i)18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.939 + 1.62i)3-s + (−0.499 − 0.866i)4-s + (0.173 − 0.300i)5-s + 1.87·6-s + (0.939 − 0.342i)7-s − 0.999·8-s + (−1.26 + 2.19i)9-s + (−0.173 − 0.300i)10-s + (0.939 − 1.62i)12-s − 13-s + (0.173 − 0.984i)14-s + 0.652·15-s + (−0.5 + 0.866i)16-s + (−0.766 − 1.32i)17-s + (1.26 + 2.19i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.993 + 0.110i$
Analytic conductor: \(0.363319\)
Root analytic conductor: \(0.602759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :0),\ 0.993 + 0.110i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.546912433\)
\(L(\frac12)\) \(\approx\) \(1.546912433\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + T \)
good3 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.347T + T^{2} \)
47 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.53T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.50116412773392212766199792039, −9.818999160227584418831701176768, −9.140190523670367150178835979769, −8.508314788634195102090187032364, −7.29278862117808698314876138238, −5.35876320235666022761637664214, −4.85709306369948979757083774324, −4.19740199032685333669106773632, −3.07802686664497458646233500941, −2.11976201716106023149395748836, 1.92745872459092104670714350459, 2.83068459442663857027014777382, 4.19461207490132272649139514298, 5.54860190911144670133000919322, 6.43223368239240947867521119310, 7.21859463632192171769740235408, 7.81536478865275433794450362138, 8.620380122752069587593775639819, 9.117524890587574854919453485725, 10.75051137707483077226880935266

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