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Properties

Label 2-728-13.3-c1-0-10
Degree $2$
Conductor $728$
Sign $0.859 + 0.511i$
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  + (−0.809 − 1.40i)3-s + 3.61·5-s + (0.5 − 0.866i)7-s + (0.190 − 0.330i)9-s + (3.04 + 5.27i)11-s + (2.5 + 2.59i)13-s + (−2.92 − 5.06i)15-s + (0.5 − 0.866i)17-s + (−1.42 + 2.47i)19-s − 1.61·21-s + (−1 − 1.73i)23-s + 8.09·25-s − 5.47·27-s + (4.04 + 7.00i)29-s − 5.47·31-s + ⋯
L(s)  = 1  + (−0.467 − 0.809i)3-s + 1.61·5-s + (0.188 − 0.327i)7-s + (0.0636 − 0.110i)9-s + (0.918 + 1.59i)11-s + (0.693 + 0.720i)13-s + (−0.755 − 1.30i)15-s + (0.121 − 0.210i)17-s + (−0.327 + 0.567i)19-s − 0.353·21-s + (−0.208 − 0.361i)23-s + 1.61·25-s − 1.05·27-s + (0.751 + 1.30i)29-s − 0.982·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83457 - 0.504202i\)
\(L(\frac12)\) \(\approx\) \(1.83457 - 0.504202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-2.5 - 2.59i)T \)
good3 \( 1 + (0.809 + 1.40i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.61T + 5T^{2} \)
11 \( 1 + (-3.04 - 5.27i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.42 - 2.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.04 - 7.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.47T + 31T^{2} \)
37 \( 1 + (4.47 + 7.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.38 - 2.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.54 + 6.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 4.70T + 53T^{2} \)
59 \( 1 + (-4.73 + 8.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.73 + 4.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.381 + 0.661i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.9T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 - 3.47T + 83T^{2} \)
89 \( 1 + (7.66 + 13.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.78 + 3.08i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.23279212729684081907774062163, −9.492833687016073923107663956630, −8.851301803088746671863799343745, −7.32824048405983880778419656785, −6.71526322753090207393582093079, −6.12401246145812703412898726863, −5.07126457549553641160433710804, −3.89251908110043507973955232966, −1.95505813919478966661102006145, −1.48816505656350101121297038383, 1.36181943415995537935462169948, 2.82501426134706271355344956814, 4.08132637147262011149581912253, 5.35309087040255685343567245914, 5.83976633578829444077932715699, 6.51833995383988303248684714777, 8.158355845780471825831306302177, 8.963973677161090449426997626211, 9.682346390114896533390943589609, 10.48750008542381785768004703926

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