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Properties

Label 2-728-8.5-c1-0-12
Degree $2$
Conductor $728$
Sign $0.379 - 0.925i$
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  + (0.859 − 1.12i)2-s + 1.51i·3-s + (−0.522 − 1.93i)4-s + 3.25i·5-s + (1.69 + 1.29i)6-s − 7-s + (−2.61 − 1.07i)8-s + 0.713·9-s + (3.65 + 2.80i)10-s + 1.48i·11-s + (2.91 − 0.790i)12-s − i·13-s + (−0.859 + 1.12i)14-s − 4.92·15-s + (−3.45 + 2.01i)16-s + 0.428·17-s + ⋯
L(s)  = 1  + (0.607 − 0.794i)2-s + 0.873i·3-s + (−0.261 − 0.965i)4-s + 1.45i·5-s + (0.693 + 0.530i)6-s − 0.377·7-s + (−0.925 − 0.379i)8-s + 0.237·9-s + (1.15 + 0.885i)10-s + 0.447i·11-s + (0.842 − 0.228i)12-s − 0.277i·13-s + (−0.229 + 0.300i)14-s − 1.27·15-s + (−0.863 + 0.504i)16-s + 0.103·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.379 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42930 + 0.959026i\)
\(L(\frac12)\) \(\approx\) \(1.42930 + 0.959026i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.859 + 1.12i)T \)
7 \( 1 + T \)
13 \( 1 + iT \)
good3 \( 1 - 1.51iT - 3T^{2} \)
5 \( 1 - 3.25iT - 5T^{2} \)
11 \( 1 - 1.48iT - 11T^{2} \)
17 \( 1 - 0.428T + 17T^{2} \)
19 \( 1 - 7.20iT - 19T^{2} \)
23 \( 1 + 4.51T + 23T^{2} \)
29 \( 1 - 9.13iT - 29T^{2} \)
31 \( 1 - 4.52T + 31T^{2} \)
37 \( 1 + 8.32iT - 37T^{2} \)
41 \( 1 - 4.08T + 41T^{2} \)
43 \( 1 - 5.99iT - 43T^{2} \)
47 \( 1 + 4.37T + 47T^{2} \)
53 \( 1 - 2.68iT - 53T^{2} \)
59 \( 1 + 10.6iT - 59T^{2} \)
61 \( 1 + 13.4iT - 61T^{2} \)
67 \( 1 + 15.0iT - 67T^{2} \)
71 \( 1 - 2.10T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 0.164T + 79T^{2} \)
83 \( 1 - 13.7iT - 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−10.71711464842190716962980753787, −9.859384409550475389430895351720, −9.526103006406654660243924421063, −7.942219767322897005366717888146, −6.78564225206152245875797743819, −6.01480283227323692528043908212, −4.90726797240514952785126595862, −3.75173140962349283981137789218, −3.29274017514775541986237506663, −1.96582166195199498271558553219, 0.73888463374471354713493901591, 2.47662743957036572640009454026, 4.09128770903861517363343198708, 4.80493874291271504803604847136, 5.89278812006538972269332067590, 6.62998662659343163451304824919, 7.58057324158697240444307960846, 8.348002608126928914510005254289, 9.016247075850291151386723090937, 9.966376600122679984610754182122

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