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Properties

Label 2-4140-5.4-c1-0-2
Degree 22
Conductor 41404140
Sign −0.983−0.181i-0.983 - 0.181i
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
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  + (0.405 − 2.19i)5-s + 3.23i·7-s − 1.54·11-s + 6.47i·13-s − 7.55i·17-s + 1.20·19-s − i·23-s + (−4.67 − 1.78i)25-s − 1.14·29-s − 6.97·31-s + (7.10 + 1.31i)35-s + 5.33i·37-s − 7.35·41-s + 3.63i·43-s + 10.3i·47-s + ⋯
L(s)  = 1  + (0.181 − 0.983i)5-s + 1.22i·7-s − 0.464·11-s + 1.79i·13-s − 1.83i·17-s + 0.277·19-s − 0.208i·23-s + (−0.934 − 0.356i)25-s − 0.211·29-s − 1.25·31-s + (1.20 + 0.221i)35-s + 0.877i·37-s − 1.14·41-s + 0.554i·43-s + 1.50i·47-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=((−0.983−0.181i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=((−0.983−0.181i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: −0.983−0.181i-0.983 - 0.181i
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4140(829,⋅)\chi_{4140} (829, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4140, ( :1/2), −0.983−0.181i)(2,\ 4140,\ (\ :1/2),\ -0.983 - 0.181i)

Particular Values

L(1)L(1) ≈\approx 0.19426365900.1942636590
L(12)L(\frac12) ≈\approx 0.19426365900.1942636590
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
5 1+(−0.405+2.19i)T 1 + (-0.405 + 2.19i)T
23 1+iT 1 + iT
good7 1−3.23iT−7T2 1 - 3.23iT - 7T^{2}
11 1+1.54T+11T2 1 + 1.54T + 11T^{2}
13 1−6.47iT−13T2 1 - 6.47iT - 13T^{2}
17 1+7.55iT−17T2 1 + 7.55iT - 17T^{2}
19 1−1.20T+19T2 1 - 1.20T + 19T^{2}
29 1+1.14T+29T2 1 + 1.14T + 29T^{2}
31 1+6.97T+31T2 1 + 6.97T + 31T^{2}
37 1−5.33iT−37T2 1 - 5.33iT - 37T^{2}
41 1+7.35T+41T2 1 + 7.35T + 41T^{2}
43 1−3.63iT−43T2 1 - 3.63iT - 43T^{2}
47 1−10.3iT−47T2 1 - 10.3iT - 47T^{2}
53 1+3.16iT−53T2 1 + 3.16iT - 53T^{2}
59 1+8.15T+59T2 1 + 8.15T + 59T^{2}
61 1−0.160T+61T2 1 - 0.160T + 61T^{2}
67 1+15.1iT−67T2 1 + 15.1iT - 67T^{2}
71 1+3.24T+71T2 1 + 3.24T + 71T^{2}
73 1+9.51iT−73T2 1 + 9.51iT - 73T^{2}
79 1−10.9T+79T2 1 - 10.9T + 79T^{2}
83 1+3.26iT−83T2 1 + 3.26iT - 83T^{2}
89 1+15.5T+89T2 1 + 15.5T + 89T^{2}
97 1+9.21iT−97T2 1 + 9.21iT - 97T^{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

−8.968796920593463773742191862424, −8.150085270756566031936499038280, −7.33553038268707147529037586111, −6.50080297488980830238667419498, −5.71454321587756098142714416372, −4.92044003766219723342243504457, −4.58600037916135564557636902474, −3.26454042124610157914849916071, −2.30625722223189211123695811689, −1.52779791736151002113205937555, 0.05226301547639768587862429367, 1.44667244696725830634984769924, 2.58306346433145803124600808933, 3.61594372593802534436942129512, 3.87188710127314706725863691943, 5.33967926820519200830979682489, 5.76603236744414168855960955760, 6.74284729441168765066622902030, 7.38645536284637660464929951761, 7.904049003364759680203528944413

Graph of the ZZ-function along the critical line