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Properties

Label 2-84-12.11-c5-0-12
Degree 22
Conductor 8484
Sign −0.781−0.624i-0.781 - 0.624i
Analytic cond. 13.472213.4722
Root an. cond. 3.670453.67045
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.35 + 1.82i)2-s + (−15.1 + 3.68i)3-s + (25.3 − 19.5i)4-s + 66.1i·5-s + (74.3 − 47.4i)6-s − 49i·7-s + (−99.7 + 151. i)8-s + (215. − 111. i)9-s + (−120. − 354. i)10-s + 653.·11-s + (−311. + 389. i)12-s + 260.·13-s + (89.5 + 262. i)14-s + (−243. − 1.00e3i)15-s + (257. − 991. i)16-s − 5.25i·17-s + ⋯
L(s)  = 1  + (−0.946 + 0.323i)2-s + (−0.971 + 0.236i)3-s + (0.791 − 0.611i)4-s + 1.18i·5-s + (0.843 − 0.537i)6-s − 0.377i·7-s + (−0.551 + 0.834i)8-s + (0.888 − 0.459i)9-s + (−0.382 − 1.11i)10-s + 1.62·11-s + (−0.624 + 0.781i)12-s + 0.427·13-s + (0.122 + 0.357i)14-s + (−0.279 − 1.14i)15-s + (0.251 − 0.967i)16-s − 0.00441i·17-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((−0.781−0.624i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=((−0.781−0.624i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: −0.781−0.624i-0.781 - 0.624i
Analytic conductor: 13.472213.4722
Root analytic conductor: 3.670453.67045
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ84(71,⋅)\chi_{84} (71, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :5/2), −0.781−0.624i)(2,\ 84,\ (\ :5/2),\ -0.781 - 0.624i)

Particular Values

L(3)L(3) ≈\approx 0.223199+0.637079i0.223199 + 0.637079i
L(12)L(\frac12) ≈\approx 0.223199+0.637079i0.223199 + 0.637079i
L(72)L(\frac{7}{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+(5.35−1.82i)T 1 + (5.35 - 1.82i)T
3 1+(15.1−3.68i)T 1 + (15.1 - 3.68i)T
7 1+49iT 1 + 49iT
good5 1−66.1iT−3.12e3T2 1 - 66.1iT - 3.12e3T^{2}
11 1−653.T+1.61e5T2 1 - 653.T + 1.61e5T^{2}
13 1−260.T+3.71e5T2 1 - 260.T + 3.71e5T^{2}
17 1+5.25iT−1.41e6T2 1 + 5.25iT - 1.41e6T^{2}
19 1−2.67e3iT−2.47e6T2 1 - 2.67e3iT - 2.47e6T^{2}
23 1+4.64e3T+6.43e6T2 1 + 4.64e3T + 6.43e6T^{2}
29 1+1.16e3iT−2.05e7T2 1 + 1.16e3iT - 2.05e7T^{2}
31 1+281.iT−2.86e7T2 1 + 281. iT - 2.86e7T^{2}
37 1+1.87e3T+6.93e7T2 1 + 1.87e3T + 6.93e7T^{2}
41 1−1.25e4iT−1.15e8T2 1 - 1.25e4iT - 1.15e8T^{2}
43 1−1.81e4iT−1.47e8T2 1 - 1.81e4iT - 1.47e8T^{2}
47 1−9.83e3T+2.29e8T2 1 - 9.83e3T + 2.29e8T^{2}
53 1+2.71e4iT−4.18e8T2 1 + 2.71e4iT - 4.18e8T^{2}
59 1+2.03e4T+7.14e8T2 1 + 2.03e4T + 7.14e8T^{2}
61 1+8.68e3T+8.44e8T2 1 + 8.68e3T + 8.44e8T^{2}
67 1+6.10e3iT−1.35e9T2 1 + 6.10e3iT - 1.35e9T^{2}
71 1−1.74e4T+1.80e9T2 1 - 1.74e4T + 1.80e9T^{2}
73 1+6.84e4T+2.07e9T2 1 + 6.84e4T + 2.07e9T^{2}
79 1−7.56e4iT−3.07e9T2 1 - 7.56e4iT - 3.07e9T^{2}
83 1−5.85e4T+3.93e9T2 1 - 5.85e4T + 3.93e9T^{2}
89 1−3.02e4iT−5.58e9T2 1 - 3.02e4iT - 5.58e9T^{2}
97 1+7.48e4T+8.58e9T2 1 + 7.48e4T + 8.58e9T^{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

−14.18170156957362024384319735577, −12.07696275826593872474542407312, −11.32539046700915773402133183482, −10.34692321339590409430170336364, −9.621783976373133757667499667566, −7.88735840792377773766398475859, −6.58117344759527883063769874338, −6.08274393259955855526905428455, −3.83529925293942552130139127711, −1.43631814534852605663930157851, 0.47922271355392630691099685477, 1.66483469869168392189802234248, 4.23568271351250942067964139512, 5.91157050429856217844286070332, 7.08850811491920105262065544511, 8.648285314143350976447873288491, 9.367101803730957443362329687043, 10.76955176011419992309372930082, 11.94626072260756352508103150891, 12.25182473042320115552260739797

Graph of the ZZ-function along the critical line