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Properties

Label 2-4284-17.4-c1-0-2
Degree 22
Conductor 42844284
Sign −0.989+0.143i-0.989 + 0.143i
Analytic cond. 34.207934.2079
Root an. cond. 5.848755.84875
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  + (−1.25 + 1.25i)5-s + (0.707 + 0.707i)7-s + (2.12 + 2.12i)11-s − 2.59·13-s + (−1.08 − 3.97i)17-s + 4.57i·19-s + (−3.09 − 3.09i)23-s + 1.86i·25-s + (−1.70 + 1.70i)29-s + (1.86 − 1.86i)31-s − 1.77·35-s + (0.372 − 0.372i)37-s + (8.52 + 8.52i)41-s + 2.63i·43-s − 9.67·47-s + ⋯
L(s)  = 1  + (−0.560 + 0.560i)5-s + (0.267 + 0.267i)7-s + (0.639 + 0.639i)11-s − 0.720·13-s + (−0.263 − 0.964i)17-s + 1.04i·19-s + (−0.644 − 0.644i)23-s + 0.372i·25-s + (−0.316 + 0.316i)29-s + (0.335 − 0.335i)31-s − 0.299·35-s + (0.0612 − 0.0612i)37-s + (1.33 + 1.33i)41-s + 0.402i·43-s − 1.41·47-s + ⋯

Functional equation

Λ(s)=(4284s/2 ΓC(s) L(s)=((−0.989+0.143i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4284s/2 ΓC(s+1/2) L(s)=((−0.989+0.143i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 42844284    =    22⋅32⋅7⋅172^{2} \cdot 3^{2} \cdot 7 \cdot 17
Sign: −0.989+0.143i-0.989 + 0.143i
Analytic conductor: 34.207934.2079
Root analytic conductor: 5.848755.84875
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4284(4033,⋅)\chi_{4284} (4033, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4284, ( :1/2), −0.989+0.143i)(2,\ 4284,\ (\ :1/2),\ -0.989 + 0.143i)

Particular Values

L(1)L(1) ≈\approx 0.41225733890.4122573389
L(12)L(\frac12) ≈\approx 0.41225733890.4122573389
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
3 1 1
7 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
17 1+(1.08+3.97i)T 1 + (1.08 + 3.97i)T
good5 1+(1.25−1.25i)T−5iT2 1 + (1.25 - 1.25i)T - 5iT^{2}
11 1+(−2.12−2.12i)T+11iT2 1 + (-2.12 - 2.12i)T + 11iT^{2}
13 1+2.59T+13T2 1 + 2.59T + 13T^{2}
19 1−4.57iT−19T2 1 - 4.57iT - 19T^{2}
23 1+(3.09+3.09i)T+23iT2 1 + (3.09 + 3.09i)T + 23iT^{2}
29 1+(1.70−1.70i)T−29iT2 1 + (1.70 - 1.70i)T - 29iT^{2}
31 1+(−1.86+1.86i)T−31iT2 1 + (-1.86 + 1.86i)T - 31iT^{2}
37 1+(−0.372+0.372i)T−37iT2 1 + (-0.372 + 0.372i)T - 37iT^{2}
41 1+(−8.52−8.52i)T+41iT2 1 + (-8.52 - 8.52i)T + 41iT^{2}
43 1−2.63iT−43T2 1 - 2.63iT - 43T^{2}
47 1+9.67T+47T2 1 + 9.67T + 47T^{2}
53 1+3.88iT−53T2 1 + 3.88iT - 53T^{2}
59 1+2.63iT−59T2 1 + 2.63iT - 59T^{2}
61 1+(6.61+6.61i)T+61iT2 1 + (6.61 + 6.61i)T + 61iT^{2}
67 1+14.9T+67T2 1 + 14.9T + 67T^{2}
71 1+(10.1−10.1i)T−71iT2 1 + (10.1 - 10.1i)T - 71iT^{2}
73 1+(−2.22+2.22i)T−73iT2 1 + (-2.22 + 2.22i)T - 73iT^{2}
79 1+(−12.2−12.2i)T+79iT2 1 + (-12.2 - 12.2i)T + 79iT^{2}
83 1+12.6iT−83T2 1 + 12.6iT - 83T^{2}
89 1+2.05T+89T2 1 + 2.05T + 89T^{2}
97 1+(−5.08+5.08i)T−97iT2 1 + (-5.08 + 5.08i)T - 97iT^{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

−8.721258524622026226992200826183, −7.83421434268766365614337477670, −7.44318492294681857180512084483, −6.62715173944931811348381365012, −5.94987892155813546545957563547, −4.87329171354037681969327420960, −4.33110345133497116660275100775, −3.36642470547638034309603468408, −2.52240391368170323906100751277, −1.51684071757675467573098975644, 0.11904979256340121145620600998, 1.26575016074157810792368210388, 2.39310618181674495304608371329, 3.53138834227532555850093100977, 4.25246153816916240512087238170, 4.84741068933841477436032792791, 5.84432913833350252015971263355, 6.50275231233782144783337377743, 7.48814984022643412903506029917, 7.906996605391901355830194400859

Graph of the ZZ-function along the critical line