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Properties

Label 2-4140-5.4-c1-0-17
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 1.6106750301.610675030
L(12)L(\frac12) ≈\approx 1.6106750301.610675030
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

−8.872989383164409264593817299063, −8.051533148653196585923443281873, −7.23470705638931317577581981751, −6.32023612803338911279219611137, −6.18812998334913037474845765414, −5.07705953879800500778526527771, −4.01874362768357286045671369852, −3.47422371877217890140562727036, −2.26989075125737377128920439161, −1.74006149647777357861298374175, 0.53683687832155513978115138460, 1.02108185001736288278435560201, 2.55640106304028548719251033346, 3.58005386985303398275008998149, 4.23715758469268237149434553739, 5.16483531381594324791138482859, 5.58534342520136672793057461824, 6.77445153124924127909779610060, 7.60062098169137465578135988590, 7.79484677334856072527561612582

Graph of the ZZ-function along the critical line