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Properties

Label 2-728-13.10-c1-0-4
Degree $2$
Conductor $728$
Sign $-0.835 - 0.549i$
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  + (1.64 + 2.85i)3-s + 0.585i·5-s + (−0.866 − 0.5i)7-s + (−3.94 + 6.82i)9-s + (−2.18 + 1.26i)11-s + (1.94 − 3.03i)13-s + (−1.67 + 0.965i)15-s + (−3.75 + 6.50i)17-s + (−1.63 − 0.943i)19-s − 3.29i·21-s + (−1.62 − 2.81i)23-s + 4.65·25-s − 16.1·27-s + (3.51 + 6.08i)29-s + 9.37i·31-s + ⋯
L(s)  = 1  + (0.952 + 1.64i)3-s + 0.261i·5-s + (−0.327 − 0.188i)7-s + (−1.31 + 2.27i)9-s + (−0.660 + 0.381i)11-s + (0.538 − 0.842i)13-s + (−0.431 + 0.249i)15-s + (−0.910 + 1.57i)17-s + (−0.375 − 0.216i)19-s − 0.719i·21-s + (−0.338 − 0.586i)23-s + 0.931·25-s − 3.10·27-s + (0.652 + 1.12i)29-s + 1.68i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.485866 + 1.62376i\)
\(L(\frac12)\) \(\approx\) \(0.485866 + 1.62376i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-1.94 + 3.03i)T \)
good3 \( 1 + (-1.64 - 2.85i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.585iT - 5T^{2} \)
11 \( 1 + (2.18 - 1.26i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.75 - 6.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.63 + 0.943i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.62 + 2.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.51 - 6.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.37iT - 31T^{2} \)
37 \( 1 + (-9.17 + 5.29i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.38 + 2.53i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.82 + 4.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.57iT - 47T^{2} \)
53 \( 1 + 1.36T + 53T^{2} \)
59 \( 1 + (-3.16 - 1.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.38 - 7.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.03 + 0.599i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.22 + 1.28i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.75iT - 73T^{2} \)
79 \( 1 + 3.16T + 79T^{2} \)
83 \( 1 - 9.05iT - 83T^{2} \)
89 \( 1 + (-14.9 + 8.60i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.57 + 1.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.64243719811846638831649884953, −10.11872249259883880263539474358, −8.772974325948425664384698085175, −8.677156382922899718560645792830, −7.52186807221182414290614301277, −6.19278658988431875870478260496, −5.05213912871442083652445732105, −4.20115193722445611482741383346, −3.32599097478701615381895509824, −2.40920516927504064662696180432, 0.76736878506720515220479837062, 2.25133533614958140288631548779, 2.94187406757566003079031221421, 4.40444659049220948120342369831, 6.02876583444038465951317937058, 6.55739640304527770336286878727, 7.60210045063217773790814794180, 8.149206230356435770314082459369, 9.088699919559431872801918161954, 9.592823682161254889571830195598

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