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Properties

Label 2-168-168.5-c1-0-5
Degree 22
Conductor 168168
Sign −0.276−0.960i-0.276 - 0.960i
Analytic cond. 1.341481.34148
Root an. cond. 1.158221.15822
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  + (−0.998 + 1.00i)2-s + (0.390 + 1.68i)3-s + (−0.00659 − 1.99i)4-s + (1.54 + 0.894i)5-s + (−2.08 − 1.29i)6-s + (2.63 + 0.230i)7-s + (2.00 + 1.99i)8-s + (−2.69 + 1.31i)9-s + (−2.44 + 0.658i)10-s + (0.501 + 0.868i)11-s + (3.37 − 0.792i)12-s − 2.47·13-s + (−2.86 + 2.41i)14-s + (−0.904 + 2.96i)15-s + (−3.99 + 0.0263i)16-s + (−3.32 − 5.76i)17-s + ⋯
L(s)  = 1  + (−0.705 + 0.708i)2-s + (0.225 + 0.974i)3-s + (−0.00329 − 0.999i)4-s + (0.692 + 0.399i)5-s + (−0.849 − 0.528i)6-s + (0.996 + 0.0869i)7-s + (0.710 + 0.703i)8-s + (−0.898 + 0.439i)9-s + (−0.772 + 0.208i)10-s + (0.151 + 0.261i)11-s + (0.973 − 0.228i)12-s − 0.685·13-s + (−0.764 + 0.644i)14-s + (−0.233 + 0.765i)15-s + (−0.999 + 0.00659i)16-s + (−0.807 − 1.39i)17-s + ⋯

Functional equation

Λ(s)=(168s/2 ΓC(s) L(s)=((−0.276−0.960i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(168s/2 ΓC(s+1/2) L(s)=((−0.276−0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 168168    =    23⋅3⋅72^{3} \cdot 3 \cdot 7
Sign: −0.276−0.960i-0.276 - 0.960i
Analytic conductor: 1.341481.34148
Root analytic conductor: 1.158221.15822
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ168(5,⋅)\chi_{168} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 168, ( :1/2), −0.276−0.960i)(2,\ 168,\ (\ :1/2),\ -0.276 - 0.960i)

Particular Values

L(1)L(1) ≈\approx 0.597743+0.794155i0.597743 + 0.794155i
L(12)L(\frac12) ≈\approx 0.597743+0.794155i0.597743 + 0.794155i
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+(0.998−1.00i)T 1 + (0.998 - 1.00i)T
3 1+(−0.390−1.68i)T 1 + (-0.390 - 1.68i)T
7 1+(−2.63−0.230i)T 1 + (-2.63 - 0.230i)T
good5 1+(−1.54−0.894i)T+(2.5+4.33i)T2 1 + (-1.54 - 0.894i)T + (2.5 + 4.33i)T^{2}
11 1+(−0.501−0.868i)T+(−5.5+9.52i)T2 1 + (-0.501 - 0.868i)T + (-5.5 + 9.52i)T^{2}
13 1+2.47T+13T2 1 + 2.47T + 13T^{2}
17 1+(3.32+5.76i)T+(−8.5+14.7i)T2 1 + (3.32 + 5.76i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.85−3.22i)T+(−9.5−16.4i)T2 1 + (1.85 - 3.22i)T + (-9.5 - 16.4i)T^{2}
23 1+(−6.85−3.95i)T+(11.5+19.9i)T2 1 + (-6.85 - 3.95i)T + (11.5 + 19.9i)T^{2}
29 1−0.748T+29T2 1 - 0.748T + 29T^{2}
31 1+(−2.87+1.65i)T+(15.5−26.8i)T2 1 + (-2.87 + 1.65i)T + (15.5 - 26.8i)T^{2}
37 1+(−3.22−1.86i)T+(18.5+32.0i)T2 1 + (-3.22 - 1.86i)T + (18.5 + 32.0i)T^{2}
41 1−2.01T+41T2 1 - 2.01T + 41T^{2}
43 1+9.19iT−43T2 1 + 9.19iT - 43T^{2}
47 1+(−1.19+2.07i)T+(−23.5−40.7i)T2 1 + (-1.19 + 2.07i)T + (-23.5 - 40.7i)T^{2}
53 1+(6.33+10.9i)T+(−26.5+45.8i)T2 1 + (6.33 + 10.9i)T + (-26.5 + 45.8i)T^{2}
59 1+(7.34−4.24i)T+(29.5−51.0i)T2 1 + (7.34 - 4.24i)T + (29.5 - 51.0i)T^{2}
61 1+(−2.02+3.50i)T+(−30.5−52.8i)T2 1 + (-2.02 + 3.50i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.89+3.98i)T+(33.5−58.0i)T2 1 + (-6.89 + 3.98i)T + (33.5 - 58.0i)T^{2}
71 1−5.46iT−71T2 1 - 5.46iT - 71T^{2}
73 1+(5.68−3.28i)T+(36.5−63.2i)T2 1 + (5.68 - 3.28i)T + (36.5 - 63.2i)T^{2}
79 1+(−2.53+4.39i)T+(−39.5−68.4i)T2 1 + (-2.53 + 4.39i)T + (-39.5 - 68.4i)T^{2}
83 1−5.65iT−83T2 1 - 5.65iT - 83T^{2}
89 1+(7.39−12.8i)T+(−44.5−77.0i)T2 1 + (7.39 - 12.8i)T + (-44.5 - 77.0i)T^{2}
97 1−1.75iT−97T2 1 - 1.75iT - 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

−13.64043823237657268490785298657, −11.62882145215073466512596254091, −10.80519573696726124032244652748, −9.844629036107436029080113886853, −9.163875623546569544570273483708, −8.080652925622285221583954032222, −6.90680291842493838146868094879, −5.48057274465507297595271789518, −4.64021867845192934728007096028, −2.32726957215975913717354084240, 1.37570155427610002387879104964, 2.58789442981405495727976601237, 4.62441754964436348600338499754, 6.36104563841320582265555774112, 7.57423283467712565948956298195, 8.561268008739580609431112448696, 9.206951885531377592236420494069, 10.70374857725386098805850008449, 11.40527829911559607031112008201, 12.62982350559623117745019201084

Graph of the ZZ-function along the critical line