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Properties

Label 2-728-56.27-c1-0-35
Degree $2$
Conductor $728$
Sign $0.896 - 0.443i$
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  + (−1.32 + 0.488i)2-s − 0.610i·3-s + (1.52 − 1.29i)4-s + 0.780·5-s + (0.298 + 0.810i)6-s + (2.18 + 1.49i)7-s + (−1.38 + 2.46i)8-s + 2.62·9-s + (−1.03 + 0.381i)10-s + 3.13·11-s + (−0.791 − 0.930i)12-s − 13-s + (−3.62 − 0.911i)14-s − 0.476i·15-s + (0.638 − 3.94i)16-s + 4.47i·17-s + ⋯
L(s)  = 1  + (−0.938 + 0.345i)2-s − 0.352i·3-s + (0.761 − 0.648i)4-s + 0.349·5-s + (0.121 + 0.330i)6-s + (0.826 + 0.563i)7-s + (−0.490 + 0.871i)8-s + 0.875·9-s + (−0.327 + 0.120i)10-s + 0.945·11-s + (−0.228 − 0.268i)12-s − 0.277·13-s + (−0.969 − 0.243i)14-s − 0.123i·15-s + (0.159 − 0.987i)16-s + 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.896 - 0.443i$
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: \((2,\ 728,\ (\ :1/2),\ 0.896 - 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24939 + 0.291932i\)
\(L(\frac12)\) \(\approx\) \(1.24939 + 0.291932i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.488i)T \)
7 \( 1 + (-2.18 - 1.49i)T \)
13 \( 1 + T \)
good3 \( 1 + 0.610iT - 3T^{2} \)
5 \( 1 - 0.780T + 5T^{2} \)
11 \( 1 - 3.13T + 11T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 + 1.92iT - 19T^{2} \)
23 \( 1 - 5.54iT - 23T^{2} \)
29 \( 1 - 2.44iT - 29T^{2} \)
31 \( 1 + 5.76T + 31T^{2} \)
37 \( 1 + 0.0497iT - 37T^{2} \)
41 \( 1 + 9.71iT - 41T^{2} \)
43 \( 1 - 9.34T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 - 7.60iT - 53T^{2} \)
59 \( 1 + 13.2iT - 59T^{2} \)
61 \( 1 + 8.35T + 61T^{2} \)
67 \( 1 - 0.937T + 67T^{2} \)
71 \( 1 - 13.0iT - 71T^{2} \)
73 \( 1 + 0.406iT - 73T^{2} \)
79 \( 1 + 4.03iT - 79T^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 + 7.10iT - 97T^{2} \)
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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

−10.38097378086689535095053652566, −9.324364663841997554849060491162, −8.919574850484758893712775858713, −7.74457580879141515712422398908, −7.23187659736837939713491831784, −6.14452166363986178383814541991, −5.41571825322574878303680704248, −4.01543031015067788087011844780, −2.13844835634905049001987585583, −1.39393997008968203407496665320, 1.09008569976789342109555192927, 2.25795224159362521453795518726, 3.81050115011117076632517394904, 4.61524920939532607711538879815, 6.09581174978756845626170973245, 7.17920533952731191251283526595, 7.69822041928252797913426865456, 8.871263625679487284270152982006, 9.538673104205090997051340605192, 10.24317321008934086503842079961

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