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Properties

Label 2-728-104.101-c1-0-34
Degree 22
Conductor 728728
Sign −0.277−0.960i-0.277 - 0.960i
Analytic cond. 5.813105.81310
Root an. cond. 2.411032.41103
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.105 + 1.41i)2-s + (0.380 − 0.219i)3-s + (−1.97 + 0.296i)4-s + 1.63·5-s + (0.349 + 0.513i)6-s + (0.866 + 0.5i)7-s + (−0.625 − 2.75i)8-s + (−1.40 + 2.43i)9-s + (0.171 + 2.30i)10-s + (0.958 + 1.65i)11-s + (−0.687 + 0.547i)12-s + (3.59 − 0.256i)13-s + (−0.614 + 1.27i)14-s + (0.620 − 0.358i)15-s + (3.82 − 1.17i)16-s + (2.70 − 4.68i)17-s + ⋯
L(s)  = 1  + (0.0742 + 0.997i)2-s + (0.219 − 0.126i)3-s + (−0.988 + 0.148i)4-s + 0.729·5-s + (0.142 + 0.209i)6-s + (0.327 + 0.188i)7-s + (−0.221 − 0.975i)8-s + (−0.467 + 0.810i)9-s + (0.0541 + 0.727i)10-s + (0.288 + 0.500i)11-s + (−0.198 + 0.157i)12-s + (0.997 − 0.0711i)13-s + (−0.164 + 0.340i)14-s + (0.160 − 0.0925i)15-s + (0.956 − 0.292i)16-s + (0.655 − 1.13i)17-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((−0.277−0.960i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s+1/2) L(s)=((−0.277−0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: −0.277−0.960i-0.277 - 0.960i
Analytic conductor: 5.813105.81310
Root analytic conductor: 2.411032.41103
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ728(309,⋅)\chi_{728} (309, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :1/2), −0.277−0.960i)(2,\ 728,\ (\ :1/2),\ -0.277 - 0.960i)

Particular Values

L(1)L(1) ≈\approx 1.06036+1.41025i1.06036 + 1.41025i
L(12)L(\frac12) ≈\approx 1.06036+1.41025i1.06036 + 1.41025i
L(32)L(\frac{3}{2}) 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+(−0.105−1.41i)T 1 + (-0.105 - 1.41i)T
7 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1+(−3.59+0.256i)T 1 + (-3.59 + 0.256i)T
good3 1+(−0.380+0.219i)T+(1.5−2.59i)T2 1 + (-0.380 + 0.219i)T + (1.5 - 2.59i)T^{2}
5 1−1.63T+5T2 1 - 1.63T + 5T^{2}
11 1+(−0.958−1.65i)T+(−5.5+9.52i)T2 1 + (-0.958 - 1.65i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2.70+4.68i)T+(−8.5−14.7i)T2 1 + (-2.70 + 4.68i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.50−2.61i)T+(−9.5−16.4i)T2 1 + (1.50 - 2.61i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.32−5.76i)T+(−11.5+19.9i)T2 1 + (-3.32 - 5.76i)T + (-11.5 + 19.9i)T^{2}
29 1+(8.30−4.79i)T+(14.5−25.1i)T2 1 + (8.30 - 4.79i)T + (14.5 - 25.1i)T^{2}
31 1+2.44iT−31T2 1 + 2.44iT - 31T^{2}
37 1+(−1.88−3.26i)T+(−18.5+32.0i)T2 1 + (-1.88 - 3.26i)T + (-18.5 + 32.0i)T^{2}
41 1+(−8.34+4.81i)T+(20.5−35.5i)T2 1 + (-8.34 + 4.81i)T + (20.5 - 35.5i)T^{2}
43 1+(−9.67−5.58i)T+(21.5+37.2i)T2 1 + (-9.67 - 5.58i)T + (21.5 + 37.2i)T^{2}
47 1+0.314iT−47T2 1 + 0.314iT - 47T^{2}
53 1−1.36iT−53T2 1 - 1.36iT - 53T^{2}
59 1+(6.13−10.6i)T+(−29.5−51.0i)T2 1 + (6.13 - 10.6i)T + (-29.5 - 51.0i)T^{2}
61 1+(7.24+4.18i)T+(30.5+52.8i)T2 1 + (7.24 + 4.18i)T + (30.5 + 52.8i)T^{2}
67 1+(1.26+2.19i)T+(−33.5+58.0i)T2 1 + (1.26 + 2.19i)T + (-33.5 + 58.0i)T^{2}
71 1+(14.4+8.33i)T+(35.5+61.4i)T2 1 + (14.4 + 8.33i)T + (35.5 + 61.4i)T^{2}
73 1+10.2iT−73T2 1 + 10.2iT - 73T^{2}
79 1+1.03T+79T2 1 + 1.03T + 79T^{2}
83 1−6.81T+83T2 1 - 6.81T + 83T^{2}
89 1+(−6.37+3.67i)T+(44.5−77.0i)T2 1 + (-6.37 + 3.67i)T + (44.5 - 77.0i)T^{2}
97 1+(−1.88−1.09i)T+(48.5+84.0i)T2 1 + (-1.88 - 1.09i)T + (48.5 + 84.0i)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

−10.51626829849645448459845612067, −9.303495620125479733653135530535, −9.083045594993872280579959401165, −7.72870599124237769305226576995, −7.45293839069229785944752306287, −5.96689884809542896898436589187, −5.59803680708167554000592933979, −4.51000500809399939231900006496, −3.19487719636791079604011945348, −1.61072610961187008456816303551, 0.990773779252789091190887081715, 2.31328237743223257518102669065, 3.52179016065166877723016607223, 4.27507596887252682977602512840, 5.74631806658168566831701518394, 6.19244632346348262133754430068, 7.87456989610557211547427925847, 8.869665613396903915556807967706, 9.201418028178339427336872342582, 10.28502159664990562527975876148

Graph of the ZZ-function along the critical line