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Properties

Label 2-728-728.517-c0-0-3
Degree 22
Conductor 728728
Sign −0.969+0.246i-0.969 + 0.246i
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  + (0.866 − 0.5i)2-s + (−0.965 − 1.67i)3-s + (0.499 − 0.866i)4-s + 0.517i·5-s + (−1.67 − 0.965i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−1.36 + 2.36i)9-s + (0.258 + 0.448i)10-s − 1.93·12-s + (−0.258 − 0.965i)13-s − 0.999·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + 2.73i·18-s + (1.22 + 0.707i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.965 − 1.67i)3-s + (0.499 − 0.866i)4-s + 0.517i·5-s + (−1.67 − 0.965i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−1.36 + 2.36i)9-s + (0.258 + 0.448i)10-s − 1.93·12-s + (−0.258 − 0.965i)13-s − 0.999·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + 2.73i·18-s + (1.22 + 0.707i)19-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((−0.969+0.246i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s) L(s)=((−0.969+0.246i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: −0.969+0.246i-0.969 + 0.246i
Analytic conductor: 0.3633190.363319
Root analytic conductor: 0.6027590.602759
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ728(517,⋅)\chi_{728} (517, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :0), −0.969+0.246i)(2,\ 728,\ (\ :0),\ -0.969 + 0.246i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.97273532520.9727353252
L(12)L(\frac12) ≈\approx 0.97273532520.9727353252
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
7 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
13 1+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
good3 1+(0.965+1.67i)T+(−0.5+0.866i)T2 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2}
5 1−0.517iT−T2 1 - 0.517iT - T^{2}
11 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
23 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+T2 1 + T^{2}
37 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
41 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+T2 1 + T^{2}
53 1−T2 1 - T^{2}
59 1+(0.448+0.258i)T+(0.5+0.866i)T2 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2}
61 1+(−0.965+1.67i)T+(−0.5−0.866i)T2 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2}
67 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)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+T2 1 + T^{2}
79 1+T2 1 + T^{2}
83 1−1.41iT−T2 1 - 1.41iT - T^{2}
89 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
97 1+(−0.5−0.866i)T2 1 + (-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

−10.58784810762524686326876202046, −9.822387998177571423636545788583, −8.039649174695979859371199647945, −7.19549683533480476332749574981, −6.57542422545948173032171684454, −5.86914294820299296504722163499, −5.03448634276592809840388836677, −3.37298153125330783713215686489, −2.39438837219290775156278920171, −0.912711557107462256778664155386, 2.99317266879906967942957282813, 3.91153913587727062374076781707, 4.79794109337177889767726832972, 5.44708159964282805717245528768, 6.20537830094755422218196439131, 7.14149840591360225423978517126, 8.769495308952641826636423203483, 9.312494585354325145330862119746, 10.09590946888574478561337966873, 11.23647710904880353326443204830

Graph of the ZZ-function along the critical line