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Properties

Label 2-120-5.4-c3-0-9
Degree 22
Conductor 120120
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 7.080227.08022
Root an. cond. 2.660872.66087
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + (−10 − 5i)5-s + 10i·7-s − 9·9-s − 46·11-s + 34i·13-s + (−15 + 30i)15-s + 66i·17-s − 104·19-s + 30·21-s − 164i·23-s + (75 + 100i)25-s + 27i·27-s − 224·29-s − 72·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.894 − 0.447i)5-s + 0.539i·7-s − 0.333·9-s − 1.26·11-s + 0.725i·13-s + (−0.258 + 0.516i)15-s + 0.941i·17-s − 1.25·19-s + 0.311·21-s − 1.48i·23-s + (0.599 + 0.800i)25-s + 0.192i·27-s − 1.43·29-s − 0.417·31-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 7.080227.08022
Root analytic conductor: 2.660872.66087
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ120(49,⋅)\chi_{120} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 120, ( :3/2), −0.894−0.447i)(2,\ 120,\ (\ :3/2),\ -0.894 - 0.447i)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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+3iT 1 + 3iT
5 1+(10+5i)T 1 + (10 + 5i)T
good7 1−10iT−343T2 1 - 10iT - 343T^{2}
11 1+46T+1.33e3T2 1 + 46T + 1.33e3T^{2}
13 1−34iT−2.19e3T2 1 - 34iT - 2.19e3T^{2}
17 1−66iT−4.91e3T2 1 - 66iT - 4.91e3T^{2}
19 1+104T+6.85e3T2 1 + 104T + 6.85e3T^{2}
23 1+164iT−1.21e4T2 1 + 164iT - 1.21e4T^{2}
29 1+224T+2.43e4T2 1 + 224T + 2.43e4T^{2}
31 1+72T+2.97e4T2 1 + 72T + 2.97e4T^{2}
37 1+22iT−5.06e4T2 1 + 22iT - 5.06e4T^{2}
41 1−194T+6.89e4T2 1 - 194T + 6.89e4T^{2}
43 1+108iT−7.95e4T2 1 + 108iT - 7.95e4T^{2}
47 1+480iT−1.03e5T2 1 + 480iT - 1.03e5T^{2}
53 1+286iT−1.48e5T2 1 + 286iT - 1.48e5T^{2}
59 1+426T+2.05e5T2 1 + 426T + 2.05e5T^{2}
61 1−698T+2.26e5T2 1 - 698T + 2.26e5T^{2}
67 1−328iT−3.00e5T2 1 - 328iT - 3.00e5T^{2}
71 1−188T+3.57e5T2 1 - 188T + 3.57e5T^{2}
73 1−740iT−3.89e5T2 1 - 740iT - 3.89e5T^{2}
79 1+1.16e3T+4.93e5T2 1 + 1.16e3T + 4.93e5T^{2}
83 1+412iT−5.71e5T2 1 + 412iT - 5.71e5T^{2}
89 1+1.20e3T+7.04e5T2 1 + 1.20e3T + 7.04e5T^{2}
97 1+1.38e3iT−9.12e5T2 1 + 1.38e3iT - 9.12e5T^{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

−12.65682500324660206166449716274, −11.48940014617112979277166926438, −10.51552337640630528027926206114, −8.809327550622403946182273586564, −8.166901377964906849844234471551, −6.95631103962713017950960523055, −5.55531328262322641620521993255, −4.11131109584874450519352108659, −2.23215940033862635561626329548, 0, 2.94179454888193875562430667260, 4.21276455999393334017156306189, 5.55542996319191192670459700853, 7.29610982458847357039143586029, 8.032976968043496686889291818206, 9.506714390261479503630820251213, 10.70284126557561858924728749851, 11.17132808459488361828960735804, 12.57887120014640073013862625935

Graph of the ZZ-function along the critical line