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Properties

Label 2-3960-5.4-c1-0-16
Degree 22
Conductor 39603960
Sign 0.0536−0.998i0.0536 - 0.998i
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  + (2.23 + 0.119i)5-s − 0.415i·7-s − 11-s − 4i·13-s + 6.51i·17-s − 5.20·19-s + 8.54i·23-s + (4.97 + 0.535i)25-s + 0.895·29-s − 6.73·31-s + (0.0498 − 0.928i)35-s + 8.96i·37-s − 10.0·41-s + 4.78i·43-s − 5.61i·47-s + ⋯
L(s)  = 1  + (0.998 + 0.0536i)5-s − 0.157i·7-s − 0.301·11-s − 1.10i·13-s + 1.58i·17-s − 1.19·19-s + 1.78i·23-s + (0.994 + 0.107i)25-s + 0.166·29-s − 1.21·31-s + (0.00843 − 0.156i)35-s + 1.47i·37-s − 1.57·41-s + 0.730i·43-s − 0.819i·47-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((0.0536−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0536 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s+1/2) L(s)=((0.0536−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0536 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.0536−0.998i0.0536 - 0.998i
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), 0.0536−0.998i)(2,\ 3960,\ (\ :1/2),\ 0.0536 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 1.6347944091.634794409
L(12)L(\frac12) ≈\approx 1.6347944091.634794409
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+(−2.23−0.119i)T 1 + (-2.23 - 0.119i)T
11 1+T 1 + T
good7 1+0.415iT−7T2 1 + 0.415iT - 7T^{2}
13 1+4iT−13T2 1 + 4iT - 13T^{2}
17 1−6.51iT−17T2 1 - 6.51iT - 17T^{2}
19 1+5.20T+19T2 1 + 5.20T + 19T^{2}
23 1−8.54iT−23T2 1 - 8.54iT - 23T^{2}
29 1−0.895T+29T2 1 - 0.895T + 29T^{2}
31 1+6.73T+31T2 1 + 6.73T + 31T^{2}
37 1−8.96iT−37T2 1 - 8.96iT - 37T^{2}
41 1+10.0T+41T2 1 + 10.0T + 41T^{2}
43 1−4.78iT−43T2 1 - 4.78iT - 43T^{2}
47 1+5.61iT−47T2 1 + 5.61iT - 47T^{2}
53 1−10.0iT−53T2 1 - 10.0iT - 53T^{2}
59 1+1.63T+59T2 1 + 1.63T + 59T^{2}
61 1−7.10T+61T2 1 - 7.10T + 61T^{2}
67 1−10.6iT−67T2 1 - 10.6iT - 67T^{2}
71 1−6.19T+71T2 1 - 6.19T + 71T^{2}
73 1+3.16iT−73T2 1 + 3.16iT - 73T^{2}
79 1−11.2T+79T2 1 - 11.2T + 79T^{2}
83 1−16.2iT−83T2 1 - 16.2iT - 83T^{2}
89 1−9.56T+89T2 1 - 9.56T + 89T^{2}
97 1−0.591iT−97T2 1 - 0.591iT - 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

−8.522349203719269741357099414061, −8.063027510291093673192468504113, −7.10173894331921877230091625946, −6.33154047543563406544675224907, −5.65154420082021745540229587509, −5.12750095939635252067925756847, −3.95623531235698856496547328772, −3.18307526867450662254830684866, −2.11619627420082472966996040053, −1.30453562676828394749474654898, 0.44695352635021610572848223366, 2.05063792793275134268592518253, 2.37248300343948635690673821521, 3.64350898320535029479252731615, 4.71066380048728573294697544424, 5.17399976338723382185555381468, 6.19302297176091863006783623525, 6.71909097768333334821920630110, 7.38928328018178246893432168679, 8.562430904056195594316314626142

Graph of the ZZ-function along the critical line