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Properties

Label 2-120-120.59-c1-0-18
Degree 22
Conductor 120120
Sign −0.112+0.993i-0.112 + 0.993i
Analytic cond. 0.9582040.958204
Root an. cond. 0.9788790.978879
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  + (0.541 − 1.30i)2-s + (1.30 − 1.13i)3-s + (−1.41 − 1.41i)4-s + (−2.10 + 0.765i)5-s + (−0.778 − 2.32i)6-s + 2.27·7-s + (−2.61 + 1.08i)8-s + (0.414 − 2.97i)9-s + (−0.137 + 3.15i)10-s + 4.20i·11-s + (−3.45 − 0.239i)12-s + 3.21·13-s + (1.23 − 2.97i)14-s + (−1.87 + 3.38i)15-s + 4i·16-s + 1.53·17-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (0.754 − 0.656i)3-s + (−0.707 − 0.707i)4-s + (−0.939 + 0.342i)5-s + (−0.317 − 0.948i)6-s + 0.859·7-s + (−0.923 + 0.382i)8-s + (0.138 − 0.990i)9-s + (−0.0433 + 0.999i)10-s + 1.26i·11-s + (−0.997 − 0.0692i)12-s + 0.891·13-s + (0.328 − 0.794i)14-s + (−0.484 + 0.875i)15-s + i·16-s + 0.371·17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((−0.112+0.993i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+1/2) L(s)=((−0.112+0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: −0.112+0.993i-0.112 + 0.993i
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ120(59,⋅)\chi_{120} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 120, ( :1/2), −0.112+0.993i)(2,\ 120,\ (\ :1/2),\ -0.112 + 0.993i)

Particular Values

L(1)L(1) ≈\approx 0.902873−1.01073i0.902873 - 1.01073i
L(12)L(\frac12) ≈\approx 0.902873−1.01073i0.902873 - 1.01073i
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+(−0.541+1.30i)T 1 + (-0.541 + 1.30i)T
3 1+(−1.30+1.13i)T 1 + (-1.30 + 1.13i)T
5 1+(2.10−0.765i)T 1 + (2.10 - 0.765i)T
good7 1−2.27T+7T2 1 - 2.27T + 7T^{2}
11 1−4.20iT−11T2 1 - 4.20iT - 11T^{2}
13 1−3.21T+13T2 1 - 3.21T + 13T^{2}
17 1−1.53T+17T2 1 - 1.53T + 17T^{2}
19 1+4.82T+19T2 1 + 4.82T + 19T^{2}
23 1+1.08iT−23T2 1 + 1.08iT - 23T^{2}
29 1+1.74T+29T2 1 + 1.74T + 29T^{2}
31 1−6.82iT−31T2 1 - 6.82iT - 31T^{2}
37 1+7.76T+37T2 1 + 7.76T + 37T^{2}
41 1+2.46iT−41T2 1 + 2.46iT - 41T^{2}
43 1−8.70iT−43T2 1 - 8.70iT - 43T^{2}
47 1−1.08iT−47T2 1 - 1.08iT - 47T^{2}
53 1+11.0iT−53T2 1 + 11.0iT - 53T^{2}
59 1+4.20iT−59T2 1 + 4.20iT - 59T^{2}
61 1+8.48iT−61T2 1 + 8.48iT - 61T^{2}
67 1−2.27iT−67T2 1 - 2.27iT - 67T^{2}
71 1+11.8T+71T2 1 + 11.8T + 71T^{2}
73 1+4.54iT−73T2 1 + 4.54iT - 73T^{2}
79 1−0.485iT−79T2 1 - 0.485iT - 79T^{2}
83 1−6.94T+83T2 1 - 6.94T + 83T^{2}
89 1−8.40iT−89T2 1 - 8.40iT - 89T^{2}
97 1+10.9iT−97T2 1 + 10.9iT - 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

−12.97216082706330350974783086286, −12.27362537179254073488571612798, −11.34142946486021755352006694622, −10.29893566510207950517566437808, −8.843976315282772390421537538892, −7.982847446017719924934080919433, −6.66019287383440306153192978620, −4.65946841741135282302873134696, −3.46866357297740168708394365089, −1.82041611211165762078027645365, 3.48502519311608542932796251612, 4.41687383026674587875445185159, 5.72492333139314635187667247031, 7.52931602237207382503055771970, 8.431243802206786016026014049385, 8.867759310216032457774283179868, 10.71329372075851512599534327716, 11.72680862671100206244647533593, 13.15249246524076402143237062309, 13.94802213075734169727285895460

Graph of the ZZ-function along the critical line