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Properties

Label 2-728-91.9-c1-0-16
Degree 22
Conductor 728728
Sign −0.703+0.710i-0.703 + 0.710i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (−1.5 + 2.59i)5-s + (0.5 + 2.59i)7-s − 2·9-s − 11-s + (−1 − 3.46i)13-s + (1.5 − 2.59i)15-s + (1 − 1.73i)17-s − 3·19-s + (−0.5 − 2.59i)21-s + (−2 − 3.46i)25-s + 5·27-s + (4.5 − 7.79i)29-s + (−0.5 − 0.866i)31-s + 33-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.670 + 1.16i)5-s + (0.188 + 0.981i)7-s − 0.666·9-s − 0.301·11-s + (−0.277 − 0.960i)13-s + (0.387 − 0.670i)15-s + (0.242 − 0.420i)17-s − 0.688·19-s + (−0.109 − 0.566i)21-s + (−0.400 − 0.692i)25-s + 0.962·27-s + (0.835 − 1.44i)29-s + (−0.0898 − 0.155i)31-s + 0.174·33-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((−0.703+0.710i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s+1/2) L(s)=((−0.703+0.710i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: −0.703+0.710i-0.703 + 0.710i
Analytic conductor: 5.813105.81310
Root analytic conductor: 2.411032.41103
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ728(9,⋅)\chi_{728} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 728, ( :1/2), −0.703+0.710i)(2,\ 728,\ (\ :1/2),\ -0.703 + 0.710i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1
7 1+(−0.5−2.59i)T 1 + (-0.5 - 2.59i)T
13 1+(1+3.46i)T 1 + (1 + 3.46i)T
good3 1+T+3T2 1 + T + 3T^{2}
5 1+(1.5−2.59i)T+(−2.5−4.33i)T2 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2}
11 1+T+11T2 1 + T + 11T^{2}
17 1+(−1+1.73i)T+(−8.5−14.7i)T2 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2}
19 1+3T+19T2 1 + 3T + 19T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1+(−4.5+7.79i)T+(−14.5−25.1i)T2 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2}
31 1+(0.5+0.866i)T+(−15.5+26.8i)T2 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2}
37 1+(5+8.66i)T+(−18.5+32.0i)T2 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.5−2.59i)T+(−20.5−35.5i)T2 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.5+2.59i)T+(−21.5+37.2i)T2 1 + (1.5 + 2.59i)T + (-21.5 + 37.2i)T^{2}
47 1+(5.5−9.52i)T+(−23.5−40.7i)T2 1 + (5.5 - 9.52i)T + (-23.5 - 40.7i)T^{2}
53 1+(1.5+2.59i)T+(−26.5+45.8i)T2 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2}
59 1+(6−10.3i)T+(−29.5−51.0i)T2 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2}
61 1+5T+61T2 1 + 5T + 61T^{2}
67 1+9T+67T2 1 + 9T + 67T^{2}
71 1+(−0.5−0.866i)T+(−35.5+61.4i)T2 1 + (-0.5 - 0.866i)T + (-35.5 + 61.4i)T^{2}
73 1+(5.5+9.52i)T+(−36.5+63.2i)T2 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2}
79 1+(−4.5+7.79i)T+(−39.5−68.4i)T2 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2}
83 1+8T+83T2 1 + 8T + 83T^{2}
89 1+(−5−8.66i)T+(−44.5+77.0i)T2 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2}
97 1+(−6.5−11.2i)T+(−48.5+84.0i)T2 1 + (-6.5 - 11.2i)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

−10.43314165639347719582902835866, −9.212346144177994265292567663126, −8.182539007613774978172420930244, −7.54297387751119883840500787058, −6.36867670511174030911188506353, −5.74122411813187522605003405952, −4.71576316896358109364951279152, −3.21903736313216218490155977672, −2.50341705440163373610399575474, 0, 1.45242600029716425330608173542, 3.40608782404698359156344392919, 4.60077667820527999883254340837, 5.01161062926751993932665747937, 6.34680427516949457631538116206, 7.20874132291086813603661315875, 8.362682670475705107528823623482, 8.699245434152593349296847277192, 10.00542970929772758021743164874

Graph of the ZZ-function along the critical line