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Properties

Label 2-728-13.4-c1-0-8
Degree $2$
Conductor $728$
Sign $0.992 - 0.124i$
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  + (0.138 − 0.239i)3-s − 1.57i·5-s + (−0.866 + 0.5i)7-s + (1.46 + 2.53i)9-s + (1.92 + 1.11i)11-s + (0.402 + 3.58i)13-s + (−0.377 − 0.217i)15-s + (1.57 + 2.71i)17-s + (−0.118 + 0.0682i)19-s + 0.276i·21-s + (3.69 − 6.40i)23-s + 2.51·25-s + 1.63·27-s + (3.35 − 5.80i)29-s − 1.91i·31-s + ⋯
L(s)  = 1  + (0.0798 − 0.138i)3-s − 0.704i·5-s + (−0.327 + 0.188i)7-s + (0.487 + 0.843i)9-s + (0.580 + 0.335i)11-s + (0.111 + 0.993i)13-s + (−0.0974 − 0.0562i)15-s + (0.380 + 0.659i)17-s + (−0.0271 + 0.0156i)19-s + 0.0603i·21-s + (0.771 − 1.33i)23-s + 0.503·25-s + 0.315·27-s + (0.622 − 1.07i)29-s − 0.343i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.992 - 0.124i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64159 + 0.102398i\)
\(L(\frac12)\) \(\approx\) \(1.64159 + 0.102398i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.402 - 3.58i)T \)
good3 \( 1 + (-0.138 + 0.239i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.57iT - 5T^{2} \)
11 \( 1 + (-1.92 - 1.11i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.57 - 2.71i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.118 - 0.0682i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.69 + 6.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.35 + 5.80i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.91iT - 31T^{2} \)
37 \( 1 + (-1.13 - 0.657i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.18 - 4.15i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.37 + 4.11i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.92iT - 47T^{2} \)
53 \( 1 + 1.59T + 53T^{2} \)
59 \( 1 + (8.51 - 4.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.02 - 1.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.96 + 4.02i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.58 - 0.914i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.37iT - 73T^{2} \)
79 \( 1 - 6.91T + 79T^{2} \)
83 \( 1 - 5.95iT - 83T^{2} \)
89 \( 1 + (0.981 + 0.566i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.72 + 5.03i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.35895127570895042342540346696, −9.464066398472039017852842203596, −8.738602105503720778525823430734, −7.910547596688595543434109556204, −6.87228532262721598174276024416, −6.07038342980297513900358743404, −4.75064888727601547528261136470, −4.19926103911700567445957287165, −2.56401307691343930825252445884, −1.30924576335389981730129626292, 1.06417748390597893573731676142, 3.06002738540530896809851627906, 3.53623000941561752491877861083, 4.94555011450531100410764995350, 6.07369978986131903888704677814, 6.90711447979914296060294528136, 7.58644175573355684273973831113, 8.866395668041818252617106956889, 9.524191747178062307215960980734, 10.39482120698765670806813959992

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