↑

Properties

Label 2-624-12.11-c3-0-42
Degree 22
Conductor 624624
Sign 0.954−0.297i0.954 - 0.297i
Analytic cond. 36.817136.8171
Root an. cond. 6.067716.06771
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.81 + 3.52i)3-s − 18.3i·5-s + 21.7i·7-s + (2.18 + 26.9i)9-s + 64.6·11-s − 13·13-s + (64.6 − 70.1i)15-s − 44.0i·17-s − 30.8i·19-s + (−76.7 + 83.1i)21-s + 85.6·23-s − 211.·25-s + (−86.4 + 110. i)27-s + 22.3i·29-s + 10.5i·31-s + ⋯
L(s)  = 1  + (0.735 + 0.677i)3-s − 1.64i·5-s + 1.17i·7-s + (0.0807 + 0.996i)9-s + 1.77·11-s − 0.277·13-s + (1.11 − 1.20i)15-s − 0.628i·17-s − 0.372i·19-s + (−0.797 + 0.864i)21-s + 0.776·23-s − 1.69·25-s + (−0.616 + 0.787i)27-s + 0.143i·29-s + 0.0613i·31-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=((0.954−0.297i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+3/2) L(s)=((0.954−0.297i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: 0.954−0.297i0.954 - 0.297i
Analytic conductor: 36.817136.8171
Root analytic conductor: 6.067716.06771
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ624(287,⋅)\chi_{624} (287, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 624, ( :3/2), 0.954−0.297i)(2,\ 624,\ (\ :3/2),\ 0.954 - 0.297i)

Particular Values

L(2)L(2) ≈\approx 2.9496155232.949615523
L(12)L(\frac12) ≈\approx 2.9496155232.949615523
L(52)L(\frac{5}{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+(−3.81−3.52i)T 1 + (-3.81 - 3.52i)T
13 1+13T 1 + 13T
good5 1+18.3iT−125T2 1 + 18.3iT - 125T^{2}
7 1−21.7iT−343T2 1 - 21.7iT - 343T^{2}
11 1−64.6T+1.33e3T2 1 - 64.6T + 1.33e3T^{2}
17 1+44.0iT−4.91e3T2 1 + 44.0iT - 4.91e3T^{2}
19 1+30.8iT−6.85e3T2 1 + 30.8iT - 6.85e3T^{2}
23 1−85.6T+1.21e4T2 1 - 85.6T + 1.21e4T^{2}
29 1−22.3iT−2.43e4T2 1 - 22.3iT - 2.43e4T^{2}
31 1−10.5iT−2.97e4T2 1 - 10.5iT - 2.97e4T^{2}
37 1−237.T+5.06e4T2 1 - 237.T + 5.06e4T^{2}
41 1−35.5iT−6.89e4T2 1 - 35.5iT - 6.89e4T^{2}
43 1−499.iT−7.95e4T2 1 - 499. iT - 7.95e4T^{2}
47 1+204.T+1.03e5T2 1 + 204.T + 1.03e5T^{2}
53 1+669.iT−1.48e5T2 1 + 669. iT - 1.48e5T^{2}
59 1−541.T+2.05e5T2 1 - 541.T + 2.05e5T^{2}
61 1−929.T+2.26e5T2 1 - 929.T + 2.26e5T^{2}
67 1−325.iT−3.00e5T2 1 - 325. iT - 3.00e5T^{2}
71 1−307.T+3.57e5T2 1 - 307.T + 3.57e5T^{2}
73 1−752.T+3.89e5T2 1 - 752.T + 3.89e5T^{2}
79 1+1.29e3iT−4.93e5T2 1 + 1.29e3iT - 4.93e5T^{2}
83 1−945.T+5.71e5T2 1 - 945.T + 5.71e5T^{2}
89 1−1.27e3iT−7.04e5T2 1 - 1.27e3iT - 7.04e5T^{2}
97 1+550.T+9.12e5T2 1 + 550.T + 9.12e5T^{2}
show more
show less
   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

−9.618579018934826003586827164149, −9.373982165737838246998220173124, −8.723641806960006407503920159683, −8.060788002733661112421507825382, −6.61339397906148251606596344516, −5.31429918342444675782067462574, −4.74046121353493097867489885534, −3.72488822726590339089211268457, −2.35135831127094153389794797475, −1.08654901844638948088925706074, 1.01271512624292967638980271201, 2.28193795944439274603704093679, 3.54237529857513489022102171945, 3.98922522621677361586455814106, 6.11200546867064423202084487179, 6.91070003637956589118017703295, 7.18255970003991982501442918355, 8.252562588083500255085848653711, 9.378876017087823061310725141374, 10.14583337037560467564355126476

Graph of the ZZ-function along the critical line