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Properties

Label 2-960-5.4-c1-0-3
Degree 22
Conductor 960960
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 7.665637.66563
Root an. cond. 2.768682.76868
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  + i·3-s + (2 − i)5-s + 2i·7-s − 9-s − 6·11-s + 2i·13-s + (1 + 2i)15-s + 6i·17-s − 4·19-s − 2·21-s + 8i·23-s + (3 − 4i)25-s − i·27-s + 8·31-s − 6i·33-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.894 − 0.447i)5-s + 0.755i·7-s − 0.333·9-s − 1.80·11-s + 0.554i·13-s + (0.258 + 0.516i)15-s + 1.45i·17-s − 0.917·19-s − 0.436·21-s + 1.66i·23-s + (0.600 − 0.800i)25-s − 0.192i·27-s + 1.43·31-s − 1.04i·33-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+1/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 7.665637.66563
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ960(769,⋅)\chi_{960} (769, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :1/2), −0.447−0.894i)(2,\ 960,\ (\ :1/2),\ -0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.675033+1.09222i0.675033 + 1.09222i
L(12)L(\frac12) ≈\approx 0.675033+1.09222i0.675033 + 1.09222i
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−iT 1 - iT
5 1+(−2+i)T 1 + (-2 + i)T
good7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+6T+11T2 1 + 6T + 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
17 1−6iT−17T2 1 - 6iT - 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1−8iT−23T2 1 - 8iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1+4iT−47T2 1 + 4iT - 47T^{2}
53 1+6iT−53T2 1 + 6iT - 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1−4T+71T2 1 - 4T + 71T^{2}
73 1−12iT−73T2 1 - 12iT - 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1+14T+89T2 1 + 14T + 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

−10.10843074733092975834423502054, −9.677742935051225656092075022861, −8.455995937978732418884194336711, −8.277596149553374920068614923576, −6.72118002178376890734929170176, −5.68519514885472871063497048270, −5.28413821514658668311735752097, −4.18954719814143513552709798199, −2.80070665029808225874580405786, −1.85377158794925610007949732812, 0.55855792703317023892837593089, 2.35021042146029618209966128923, 2.91049022848173918235475093563, 4.62606190514689230156777348903, 5.45498881352701941400877597056, 6.44912600717094385911057205493, 7.18008565536287803926850182462, 7.968926470881624564798648519485, 8.824488093944440135428863578877, 10.16301143950663709372774734815

Graph of the ZZ-function along the critical line