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Properties

Label 2-728-91.81-c1-0-26
Degree $2$
Conductor $728$
Sign $-0.703 - 0.710i$
Analytic cond. $5.81310$
Root an. cond. $2.41103$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (−1.5 − 2.59i)5-s + (0.5 − 2.59i)7-s − 2·9-s − 11-s + (−1 + 3.46i)13-s + (1.5 + 2.59i)15-s + (1 + 1.73i)17-s − 3·19-s + (−0.5 + 2.59i)21-s + (−2 + 3.46i)25-s + 5·27-s + (4.5 + 7.79i)29-s + (−0.5 + 0.866i)31-s + 33-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.670 − 1.16i)5-s + (0.188 − 0.981i)7-s − 0.666·9-s − 0.301·11-s + (−0.277 + 0.960i)13-s + (0.387 + 0.670i)15-s + (0.242 + 0.420i)17-s − 0.688·19-s + (−0.109 + 0.566i)21-s + (−0.400 + 0.692i)25-s + 0.962·27-s + (0.835 + 1.44i)29-s + (−0.0898 + 0.155i)31-s + 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.703 - 0.710i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.703 - 0.710i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + (1 - 3.46i)T \)
good3 \( 1 + T + 3T^{2} \)
5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + T + 11T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.5 - 2.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.5 + 9.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 + 9T + 67T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.00542970929772758021743164874, −8.699245434152593349296847277192, −8.362682670475705107528823623482, −7.20874132291086813603661315875, −6.34680427516949457631538116206, −5.01161062926751993932665747937, −4.60077667820527999883254340837, −3.40608782404698359156344392919, −1.45242600029716425330608173542, 0, 2.50341705440163373610399575474, 3.21903736313216218490155977672, 4.71576316896358109364951279152, 5.74122411813187522605003405952, 6.36867670511174030911188506353, 7.54297387751119883840500787058, 8.182539007613774978172420930244, 9.212346144177994265292567663126, 10.43314165639347719582902835866

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