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Properties

Label 2-120-120.53-c1-0-13
Degree 22
Conductor 120120
Sign 0.972+0.234i0.972 + 0.234i
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  + (1.41 + 0.0941i)2-s + (−1.68 − 0.416i)3-s + (1.98 + 0.265i)4-s + (1.62 − 1.54i)5-s + (−2.33 − 0.746i)6-s + (−0.361 + 0.361i)7-s + (2.77 + 0.561i)8-s + (2.65 + 1.40i)9-s + (2.43 − 2.02i)10-s − 2.63·11-s + (−3.22 − 1.27i)12-s + (−3.49 + 3.49i)13-s + (−0.544 + 0.476i)14-s + (−3.36 + 1.91i)15-s + (3.85 + 1.05i)16-s + (−3.61 − 3.61i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0665i)2-s + (−0.970 − 0.240i)3-s + (0.991 + 0.132i)4-s + (0.724 − 0.688i)5-s + (−0.952 − 0.304i)6-s + (−0.136 + 0.136i)7-s + (0.980 + 0.198i)8-s + (0.884 + 0.466i)9-s + (0.769 − 0.638i)10-s − 0.794·11-s + (−0.930 − 0.367i)12-s + (−0.968 + 0.968i)13-s + (−0.145 + 0.127i)14-s + (−0.869 + 0.494i)15-s + (0.964 + 0.263i)16-s + (−0.876 − 0.876i)17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.972+0.234i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+1/2) L(s)=((0.972+0.234i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.972+0.234i0.972 + 0.234i
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ120(53,⋅)\chi_{120} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 120, ( :1/2), 0.972+0.234i)(2,\ 120,\ (\ :1/2),\ 0.972 + 0.234i)

Particular Values

L(1)L(1) ≈\approx 1.47925−0.175761i1.47925 - 0.175761i
L(12)L(\frac12) ≈\approx 1.47925−0.175761i1.47925 - 0.175761i
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+(−1.41−0.0941i)T 1 + (-1.41 - 0.0941i)T
3 1+(1.68+0.416i)T 1 + (1.68 + 0.416i)T
5 1+(−1.62+1.54i)T 1 + (-1.62 + 1.54i)T
good7 1+(0.361−0.361i)T−7iT2 1 + (0.361 - 0.361i)T - 7iT^{2}
11 1+2.63T+11T2 1 + 2.63T + 11T^{2}
13 1+(3.49−3.49i)T−13iT2 1 + (3.49 - 3.49i)T - 13iT^{2}
17 1+(3.61+3.61i)T+17iT2 1 + (3.61 + 3.61i)T + 17iT^{2}
19 1−0.672T+19T2 1 - 0.672T + 19T^{2}
23 1+(4.31−4.31i)T−23iT2 1 + (4.31 - 4.31i)T - 23iT^{2}
29 1−4.76iT−29T2 1 - 4.76iT - 29T^{2}
31 1−3.73T+31T2 1 - 3.73T + 31T^{2}
37 1+(−2.82−2.82i)T+37iT2 1 + (-2.82 - 2.82i)T + 37iT^{2}
41 1+4.10iT−41T2 1 + 4.10iT - 41T^{2}
43 1+(−7.57+7.57i)T−43iT2 1 + (-7.57 + 7.57i)T - 43iT^{2}
47 1+(0.987+0.987i)T+47iT2 1 + (0.987 + 0.987i)T + 47iT^{2}
53 1+(0.646+0.646i)T+53iT2 1 + (0.646 + 0.646i)T + 53iT^{2}
59 1+4.92iT−59T2 1 + 4.92iT - 59T^{2}
61 1+6.07iT−61T2 1 + 6.07iT - 61T^{2}
67 1+(−0.349−0.349i)T+67iT2 1 + (-0.349 - 0.349i)T + 67iT^{2}
71 1+8.63iT−71T2 1 + 8.63iT - 71T^{2}
73 1+(11.3+11.3i)T+73iT2 1 + (11.3 + 11.3i)T + 73iT^{2}
79 1−4.07iT−79T2 1 - 4.07iT - 79T^{2}
83 1+(−8.53−8.53i)T+83iT2 1 + (-8.53 - 8.53i)T + 83iT^{2}
89 1−6.58T+89T2 1 - 6.58T + 89T^{2}
97 1+(0.660−0.660i)T−97iT2 1 + (0.660 - 0.660i)T - 97iT^{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

−13.39005039574373496318138410567, −12.41765741514241666190778776085, −11.76829120942313902485500131948, −10.58649482070175993340597522348, −9.445867724157522922402985093476, −7.56438374791888586325921907142, −6.47052091506511208879437123507, −5.35488946864313463135466148459, −4.58305846880449325612744215944, −2.15523670621358233049512183517, 2.56635425807087230973151481604, 4.39229959424836275096459909554, 5.65010560185608275944114146847, 6.41656676754295628925680577809, 7.64158520779358682973407520603, 10.03741051194394526131518942214, 10.45145047676257193748751156965, 11.48475266298251613884225277167, 12.68406320949440557910723493442, 13.26671572515331130111905494450

Graph of the ZZ-function along the critical line