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Properties

Label 2-3960-5.4-c1-0-42
Degree 22
Conductor 39603960
Sign 0.894+0.447i0.894 + 0.447i
Analytic cond. 31.620731.6207
Root an. cond. 5.623235.62323
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 − 2i)5-s + 2i·7-s + 11-s + 4i·17-s − 4·19-s − 6i·23-s + (−3 − 4i)25-s + 2·29-s + 8·31-s + (4 + 2i)35-s − 4i·37-s + 6·41-s + 6i·43-s + 2i·47-s + 3·49-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)5-s + 0.755i·7-s + 0.301·11-s + 0.970i·17-s − 0.917·19-s − 1.25i·23-s + (−0.600 − 0.800i)25-s + 0.371·29-s + 1.43·31-s + (0.676 + 0.338i)35-s − 0.657i·37-s + 0.937·41-s + 0.914i·43-s + 0.291i·47-s + 0.428·49-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s+1/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 31.620731.6207
Root analytic conductor: 5.623235.62323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3960(3169,⋅)\chi_{3960} (3169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :1/2), 0.894+0.447i)(2,\ 3960,\ (\ :1/2),\ 0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 2.0646125332.064612533
L(12)L(\frac12) ≈\approx 2.0646125332.064612533
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+(−1+2i)T 1 + (-1 + 2i)T
11 1−T 1 - T
good7 1−2iT−7T2 1 - 2iT - 7T^{2}
13 1−13T2 1 - 13T^{2}
17 1−4iT−17T2 1 - 4iT - 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1+4iT−37T2 1 + 4iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1−6iT−43T2 1 - 6iT - 43T^{2}
47 1−2iT−47T2 1 - 2iT - 47T^{2}
53 1+12iT−53T2 1 + 12iT - 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1−14T+61T2 1 - 14T + 61T^{2}
67 1−10iT−67T2 1 - 10iT - 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+4iT−73T2 1 + 4iT - 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1−2iT−83T2 1 - 2iT - 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+8iT−97T2 1 + 8iT - 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

−8.561690076916054853886704956183, −7.969278943331290246984175117134, −6.68899086485885057007120434305, −6.17149480136203216358665343108, −5.48925641320362647852716259597, −4.58221927067700631649930411133, −4.02440603694632319159604924134, −2.65791307822691878667918729706, −1.95780733126578720706026041785, −0.76682447496435372864109877362, 0.913959038410848158142972276161, 2.15097134920176642783220814059, 3.00571192406293325630737643907, 3.86762120371455746166277772484, 4.67497077699503002441332596121, 5.66882303551193056114112541739, 6.39804456224879840857164182770, 7.07471794438814770083035953668, 7.56380120870672226378041199914, 8.509151204991037720525387181518

Graph of the ZZ-function along the critical line