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Properties

Label 2-728-56.27-c1-0-57
Degree $2$
Conductor $728$
Sign $0.459 + 0.888i$
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  + (−0.264 + 1.38i)2-s − 1.53i·3-s + (−1.85 − 0.735i)4-s − 0.856·5-s + (2.12 + 0.405i)6-s + (−1.33 + 2.28i)7-s + (1.51 − 2.38i)8-s + 0.649·9-s + (0.226 − 1.18i)10-s − 0.850·11-s + (−1.12 + 2.85i)12-s + 13-s + (−2.82 − 2.45i)14-s + 1.31i·15-s + (2.91 + 2.73i)16-s − 5.95i·17-s + ⋯
L(s)  = 1  + (−0.187 + 0.982i)2-s − 0.885i·3-s + (−0.929 − 0.367i)4-s − 0.382·5-s + (0.869 + 0.165i)6-s + (−0.504 + 0.863i)7-s + (0.535 − 0.844i)8-s + 0.216·9-s + (0.0716 − 0.376i)10-s − 0.256·11-s + (−0.325 + 0.823i)12-s + 0.277·13-s + (−0.754 − 0.656i)14-s + 0.338i·15-s + (0.729 + 0.684i)16-s − 1.44i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.459 + 0.888i$
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: \((2,\ 728,\ (\ :1/2),\ 0.459 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.683914 - 0.416122i\)
\(L(\frac12)\) \(\approx\) \(0.683914 - 0.416122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.264 - 1.38i)T \)
7 \( 1 + (1.33 - 2.28i)T \)
13 \( 1 - T \)
good3 \( 1 + 1.53iT - 3T^{2} \)
5 \( 1 + 0.856T + 5T^{2} \)
11 \( 1 + 0.850T + 11T^{2} \)
17 \( 1 + 5.95iT - 17T^{2} \)
19 \( 1 + 3.06iT - 19T^{2} \)
23 \( 1 - 1.93iT - 23T^{2} \)
29 \( 1 + 6.57iT - 29T^{2} \)
31 \( 1 + 2.27T + 31T^{2} \)
37 \( 1 + 7.65iT - 37T^{2} \)
41 \( 1 + 2.12iT - 41T^{2} \)
43 \( 1 - 2.48T + 43T^{2} \)
47 \( 1 - 3.71T + 47T^{2} \)
53 \( 1 + 8.57iT - 53T^{2} \)
59 \( 1 + 8.59iT - 59T^{2} \)
61 \( 1 - 0.283T + 61T^{2} \)
67 \( 1 - 1.93T + 67T^{2} \)
71 \( 1 - 7.94iT - 71T^{2} \)
73 \( 1 + 3.38iT - 73T^{2} \)
79 \( 1 + 2.46iT - 79T^{2} \)
83 \( 1 - 15.7iT - 83T^{2} \)
89 \( 1 - 11.1iT - 89T^{2} \)
97 \( 1 + 8.18iT - 97T^{2} \)
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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

−9.814316340046988070716380962384, −9.294007692208010655446004896672, −8.289647369202735052572324171981, −7.50737705465163945057875906123, −6.89216784707191608697350268557, −5.98392542806952933102074877025, −5.12785396679406941558460877718, −3.86365416764677367968916541109, −2.30950232170116654098343953876, −0.47204907334760575513965608548, 1.46557661774058994003006631175, 3.24920600115340205334574485419, 3.93738505324698094792775871626, 4.59182172361865799279948164746, 5.91544883449000951295101106473, 7.29984282848419798808295042243, 8.198005567777863371319434966848, 9.092035044146208420804291377732, 10.02588922491495121665235669313, 10.47053937430857961482591999953

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