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Properties

Label 2-728-8.5-c1-0-0
Degree $2$
Conductor $728$
Sign $-0.913 + 0.406i$
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  + (−1.11 + 0.870i)2-s + 2.73i·3-s + (0.484 − 1.94i)4-s − 4.29i·5-s + (−2.37 − 3.04i)6-s − 7-s + (1.14 + 2.58i)8-s − 4.46·9-s + (3.73 + 4.78i)10-s + 3.63i·11-s + (5.30 + 1.32i)12-s − i·13-s + (1.11 − 0.870i)14-s + 11.7·15-s + (−3.53 − 1.88i)16-s − 1.14·17-s + ⋯
L(s)  = 1  + (−0.788 + 0.615i)2-s + 1.57i·3-s + (0.242 − 0.970i)4-s − 1.91i·5-s + (−0.971 − 1.24i)6-s − 0.377·7-s + (0.406 + 0.913i)8-s − 1.48·9-s + (1.18 + 1.51i)10-s + 1.09i·11-s + (1.53 + 0.382i)12-s − 0.277i·13-s + (0.297 − 0.232i)14-s + 3.02·15-s + (−0.882 − 0.470i)16-s − 0.277·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.913 + 0.406i$
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: \((2,\ 728,\ (\ :1/2),\ -0.913 + 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0706411 - 0.332733i\)
\(L(\frac12)\) \(\approx\) \(0.0706411 - 0.332733i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 - 0.870i)T \)
7 \( 1 + T \)
13 \( 1 + iT \)
good3 \( 1 - 2.73iT - 3T^{2} \)
5 \( 1 + 4.29iT - 5T^{2} \)
11 \( 1 - 3.63iT - 11T^{2} \)
17 \( 1 + 1.14T + 17T^{2} \)
19 \( 1 - 5.50iT - 19T^{2} \)
23 \( 1 + 7.34T + 23T^{2} \)
29 \( 1 - 2.77iT - 29T^{2} \)
31 \( 1 - 0.223T + 31T^{2} \)
37 \( 1 - 8.95iT - 37T^{2} \)
41 \( 1 + 1.86T + 41T^{2} \)
43 \( 1 + 0.422iT - 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + 5.01iT - 53T^{2} \)
59 \( 1 + 1.07iT - 59T^{2} \)
61 \( 1 + 4.56iT - 61T^{2} \)
67 \( 1 - 9.43iT - 67T^{2} \)
71 \( 1 + 2.96T + 71T^{2} \)
73 \( 1 - 2.96T + 73T^{2} \)
79 \( 1 - 7.12T + 79T^{2} \)
83 \( 1 - 5.14iT - 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−10.16704041191328527028705770435, −9.938405207031451035300850812279, −9.292623816515941003706433360465, −8.449669804438134756631797489853, −7.904744281086111379076655781293, −6.30020433865962328955437306944, −5.28578580409542615982909106300, −4.76800740341403685744145105914, −3.86819065769789197440140931844, −1.70107637074253058606724973601, 0.21628791616696678178261996070, 2.04743099281767413898819442187, 2.75655780961247397482001197673, 3.68271032652571630132264165483, 6.20632878860080968341944965937, 6.50794655939671511013526844888, 7.40912057917286318380265513745, 7.927601385838734047865205874080, 9.011413915065937325363841784502, 10.07715837441819387023551818264

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