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Properties

Label 2-728-56.27-c1-0-64
Degree $2$
Conductor $728$
Sign $0.730 + 0.682i$
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  + (0.906 + 1.08i)2-s − 1.31i·3-s + (−0.354 + 1.96i)4-s − 2.82·5-s + (1.42 − 1.19i)6-s + (−0.786 − 2.52i)7-s + (−2.45 + 1.39i)8-s + 1.27·9-s + (−2.56 − 3.06i)10-s + 3.42·11-s + (2.58 + 0.465i)12-s + 13-s + (2.02 − 3.14i)14-s + 3.70i·15-s + (−3.74 − 1.39i)16-s − 3.79i·17-s + ⋯
L(s)  = 1  + (0.641 + 0.767i)2-s − 0.757i·3-s + (−0.177 + 0.984i)4-s − 1.26·5-s + (0.581 − 0.486i)6-s + (−0.297 − 0.954i)7-s + (−0.868 + 0.494i)8-s + 0.425·9-s + (−0.809 − 0.968i)10-s + 1.03·11-s + (0.745 + 0.134i)12-s + 0.277·13-s + (0.542 − 0.840i)14-s + 0.957i·15-s + (−0.937 − 0.349i)16-s − 0.921i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.730 + 0.682i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.730 + 0.682i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41383 - 0.557544i\)
\(L(\frac12)\) \(\approx\) \(1.41383 - 0.557544i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.906 - 1.08i)T \)
7 \( 1 + (0.786 + 2.52i)T \)
13 \( 1 - T \)
good3 \( 1 + 1.31iT - 3T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
11 \( 1 - 3.42T + 11T^{2} \)
17 \( 1 + 3.79iT - 17T^{2} \)
19 \( 1 + 2.36iT - 19T^{2} \)
23 \( 1 + 8.79iT - 23T^{2} \)
29 \( 1 + 2.31iT - 29T^{2} \)
31 \( 1 - 5.89T + 31T^{2} \)
37 \( 1 + 1.36iT - 37T^{2} \)
41 \( 1 - 0.250iT - 41T^{2} \)
43 \( 1 + 7.71T + 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 - 8.36iT - 53T^{2} \)
59 \( 1 - 13.5iT - 59T^{2} \)
61 \( 1 + 9.02T + 61T^{2} \)
67 \( 1 + 2.61T + 67T^{2} \)
71 \( 1 - 4.45iT - 71T^{2} \)
73 \( 1 + 2.04iT - 73T^{2} \)
79 \( 1 - 0.539iT - 79T^{2} \)
83 \( 1 - 5.73iT - 83T^{2} \)
89 \( 1 + 6.16iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−10.40923797427717095370170797112, −9.096202950942069382821936841916, −8.211335693339596959526674950603, −7.37081562039949623906293342588, −6.94495850382464076799996220861, −6.20700239659605175873705312721, −4.40899385402782551674176507403, −4.21770744314606890786150583430, −2.89871362110773356080856125894, −0.69317137982342581489233853342, 1.60102511677418545097692381468, 3.42428451669198087276287479542, 3.77053424725992688455321707899, 4.75062313170815888031979154971, 5.79733416031683538912929024013, 6.78769544701681061497829148084, 8.113268745168675462800996713871, 9.084295915153068337998888918774, 9.729272982208979789314708241449, 10.62700637710813719233660646750

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