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Properties

Label 2-832-104.101-c1-0-17
Degree 22
Conductor 832832
Sign −0.759+0.650i-0.759 + 0.650i
Analytic cond. 6.643556.64355
Root an. cond. 2.577502.57750
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  + (−2.36 + 1.36i)3-s + 1.73·5-s + (−1.73 − i)7-s + (2.23 − 3.86i)9-s + (1.73 + 3i)11-s + (−3.59 − 0.232i)13-s + (−4.09 + 2.36i)15-s + (0.232 − 0.401i)17-s + (−2.36 + 4.09i)19-s + 5.46·21-s + (−2.36 − 4.09i)23-s − 2.00·25-s + 4.00i·27-s + (−0.401 + 0.232i)29-s − 0.196i·31-s + ⋯
L(s)  = 1  + (−1.36 + 0.788i)3-s + 0.774·5-s + (−0.654 − 0.377i)7-s + (0.744 − 1.28i)9-s + (0.522 + 0.904i)11-s + (−0.997 − 0.0643i)13-s + (−1.05 + 0.610i)15-s + (0.0562 − 0.0974i)17-s + (−0.542 + 0.940i)19-s + 1.19·21-s + (−0.493 − 0.854i)23-s − 0.400·25-s + 0.769i·27-s + (−0.0746 + 0.0430i)29-s − 0.0352i·31-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((−0.759+0.650i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((−0.759+0.650i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: −0.759+0.650i-0.759 + 0.650i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(673,⋅)\chi_{832} (673, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 832, ( :1/2), −0.759+0.650i)(2,\ 832,\ (\ :1/2),\ -0.759 + 0.650i)

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
13 1+(3.59+0.232i)T 1 + (3.59 + 0.232i)T
good3 1+(2.36−1.36i)T+(1.5−2.59i)T2 1 + (2.36 - 1.36i)T + (1.5 - 2.59i)T^{2}
5 1−1.73T+5T2 1 - 1.73T + 5T^{2}
7 1+(1.73+i)T+(3.5+6.06i)T2 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2}
11 1+(−1.73−3i)T+(−5.5+9.52i)T2 1 + (-1.73 - 3i)T + (-5.5 + 9.52i)T^{2}
17 1+(−0.232+0.401i)T+(−8.5−14.7i)T2 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.36−4.09i)T+(−9.5−16.4i)T2 1 + (2.36 - 4.09i)T + (-9.5 - 16.4i)T^{2}
23 1+(2.36+4.09i)T+(−11.5+19.9i)T2 1 + (2.36 + 4.09i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.401−0.232i)T+(14.5−25.1i)T2 1 + (0.401 - 0.232i)T + (14.5 - 25.1i)T^{2}
31 1+0.196iT−31T2 1 + 0.196iT - 31T^{2}
37 1+(−4.59−7.96i)T+(−18.5+32.0i)T2 1 + (-4.59 - 7.96i)T + (-18.5 + 32.0i)T^{2}
41 1+(4.5−2.59i)T+(20.5−35.5i)T2 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2}
43 1+(10.7+6.19i)T+(21.5+37.2i)T2 1 + (10.7 + 6.19i)T + (21.5 + 37.2i)T^{2}
47 1+11.6iT−47T2 1 + 11.6iT - 47T^{2}
53 1+12.4iT−53T2 1 + 12.4iT - 53T^{2}
59 1+(−0.464+0.803i)T+(−29.5−51.0i)T2 1 + (-0.464 + 0.803i)T + (-29.5 - 51.0i)T^{2}
61 1+(11.5+6.69i)T+(30.5+52.8i)T2 1 + (11.5 + 6.69i)T + (30.5 + 52.8i)T^{2}
67 1+(4.09+7.09i)T+(−33.5+58.0i)T2 1 + (4.09 + 7.09i)T + (-33.5 + 58.0i)T^{2}
71 1+(−10.0−5.83i)T+(35.5+61.4i)T2 1 + (-10.0 - 5.83i)T + (35.5 + 61.4i)T^{2}
73 1−5.19iT−73T2 1 - 5.19iT - 73T^{2}
79 1+6T+79T2 1 + 6T + 79T^{2}
83 1+2.53T+83T2 1 + 2.53T + 83T^{2}
89 1+(13.3−7.73i)T+(44.5−77.0i)T2 1 + (13.3 - 7.73i)T + (44.5 - 77.0i)T^{2}
97 1+(−5.19−3i)T+(48.5+84.0i)T2 1 + (-5.19 - 3i)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

−9.937320817230877243006730226984, −9.693092048785965799108138466402, −8.238864718995406736665597526339, −6.80090567775694803637690421120, −6.43255493432673088387957864134, −5.37965047500399563365257126725, −4.66932085531096349826805844152, −3.68258941768307417564459961712, −1.95926020103333398719218504371, 0, 1.55127966613772653635832194470, 2.86595080290625693495463666517, 4.52477829635618333634785034522, 5.72123882800760642411037093895, 6.01685940753180579280486418089, 6.83974332914137070798219456170, 7.72408930626670411408262597742, 9.075602396640237623680029351479, 9.700766675820996941323991569886

Graph of the ZZ-function along the critical line