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Properties

Label 2-728-728.555-c0-0-0
Degree 22
Conductor 728728
Sign 0.203−0.979i0.203 - 0.979i
Analytic cond. 0.3633190.363319
Root an. cond. 0.6027590.602759
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)6-s + (0.866 + 0.5i)7-s − 0.999·8-s − 0.999i·10-s + 11-s + (−0.499 + 0.866i)12-s + i·13-s + 0.999i·14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s − 19-s + (0.866 − 0.499i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)6-s + (0.866 + 0.5i)7-s − 0.999·8-s − 0.999i·10-s + 11-s + (−0.499 + 0.866i)12-s + i·13-s + 0.999i·14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s − 19-s + (0.866 − 0.499i)20-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((0.203−0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s) L(s)=((0.203−0.979i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: 0.203−0.979i0.203 - 0.979i
Analytic conductor: 0.3633190.363319
Root analytic conductor: 0.6027590.602759
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ728(555,⋅)\chi_{728} (555, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :0), 0.203−0.979i)(2,\ 728,\ (\ :0),\ 0.203 - 0.979i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4352176571.435217657
L(12)L(\frac12) ≈\approx 1.4352176571.435217657
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1−iT 1 - iT
good3 1−T+T2 1 - T + T^{2}
5 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
11 1−T+T2 1 - T + T^{2}
17 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
19 1+T+T2 1 + T + T^{2}
23 1+(−1.73+i)T+(0.5−0.866i)T2 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}
29 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
31 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
37 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
41 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
59 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
61 1+iT−T2 1 + iT - T^{2}
67 1+T+T2 1 + T + T^{2}
71 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
73 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
97 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−11.08830000986947996829912201174, −9.246718335252239253062508643782, −8.770804179891043210404216125220, −8.368409693163070938685903664242, −7.39227713002733764076852229311, −6.55890819916875520630182972208, −5.26421446667596882623117464879, −4.31870346478650307923802958900, −3.66067459010521202077898548339, −2.20973621825204242855015424542, 1.56906326891980353279485018926, 3.01903489226972674555976396679, 3.63515027289522843831417384640, 4.54839496207433494790579845691, 5.74726058659740618990277102128, 7.14145515157524413221037468650, 7.924592507317041667693927539328, 8.830446296496800449208902868975, 9.516041875471359850239043153152, 10.80903322683203537553849327876

Graph of the ZZ-function along the critical line