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Properties

Label 2-168-168.101-c1-0-1
Degree 22
Conductor 168168
Sign −0.981+0.191i-0.981 + 0.191i
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  + (−0.0857 + 1.41i)2-s + (−0.528 − 1.64i)3-s + (−1.98 − 0.242i)4-s + (−2.66 + 1.54i)5-s + (2.37 − 0.604i)6-s + (−1.46 + 2.20i)7-s + (0.511 − 2.78i)8-s + (−2.44 + 1.74i)9-s + (−1.94 − 3.89i)10-s + (0.621 − 1.07i)11-s + (0.650 + 3.40i)12-s − 5.98·13-s + (−2.98 − 2.25i)14-s + (3.95 + 3.58i)15-s + (3.88 + 0.960i)16-s + (−0.595 + 1.03i)17-s + ⋯
L(s)  = 1  + (−0.0606 + 0.998i)2-s + (−0.305 − 0.952i)3-s + (−0.992 − 0.121i)4-s + (−1.19 + 0.688i)5-s + (0.969 − 0.246i)6-s + (−0.552 + 0.833i)7-s + (0.180 − 0.983i)8-s + (−0.813 + 0.581i)9-s + (−0.615 − 1.23i)10-s + (0.187 − 0.324i)11-s + (0.187 + 0.982i)12-s − 1.66·13-s + (−0.798 − 0.602i)14-s + (1.02 + 0.926i)15-s + (0.970 + 0.240i)16-s + (−0.144 + 0.250i)17-s + ⋯

Functional equation

Λ(s)=(168s/2 ΓC(s) L(s)=((−0.981+0.191i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(168s/2 ΓC(s+1/2) L(s)=((−0.981+0.191i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 168168    =    23⋅3⋅72^{3} \cdot 3 \cdot 7
Sign: −0.981+0.191i-0.981 + 0.191i
Analytic conductor: 1.341481.34148
Root analytic conductor: 1.158221.15822
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ168(101,⋅)\chi_{168} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 168, ( :1/2), −0.981+0.191i)(2,\ 168,\ (\ :1/2),\ -0.981 + 0.191i)

Particular Values

L(1)L(1) ≈\approx 0.0214386−0.221990i0.0214386 - 0.221990i
L(12)L(\frac12) ≈\approx 0.0214386−0.221990i0.0214386 - 0.221990i
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+(0.0857−1.41i)T 1 + (0.0857 - 1.41i)T
3 1+(0.528+1.64i)T 1 + (0.528 + 1.64i)T
7 1+(1.46−2.20i)T 1 + (1.46 - 2.20i)T
good5 1+(2.66−1.54i)T+(2.5−4.33i)T2 1 + (2.66 - 1.54i)T + (2.5 - 4.33i)T^{2}
11 1+(−0.621+1.07i)T+(−5.5−9.52i)T2 1 + (-0.621 + 1.07i)T + (-5.5 - 9.52i)T^{2}
13 1+5.98T+13T2 1 + 5.98T + 13T^{2}
17 1+(0.595−1.03i)T+(−8.5−14.7i)T2 1 + (0.595 - 1.03i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.614−1.06i)T+(−9.5+16.4i)T2 1 + (-0.614 - 1.06i)T + (-9.5 + 16.4i)T^{2}
23 1+(−2.56+1.48i)T+(11.5−19.9i)T2 1 + (-2.56 + 1.48i)T + (11.5 - 19.9i)T^{2}
29 1−3.19T+29T2 1 - 3.19T + 29T^{2}
31 1+(−1.33−0.773i)T+(15.5+26.8i)T2 1 + (-1.33 - 0.773i)T + (15.5 + 26.8i)T^{2}
37 1+(0.334−0.193i)T+(18.5−32.0i)T2 1 + (0.334 - 0.193i)T + (18.5 - 32.0i)T^{2}
41 1+9.44T+41T2 1 + 9.44T + 41T^{2}
43 1−8.29iT−43T2 1 - 8.29iT - 43T^{2}
47 1+(3.34+5.78i)T+(−23.5+40.7i)T2 1 + (3.34 + 5.78i)T + (-23.5 + 40.7i)T^{2}
53 1+(5.25−9.09i)T+(−26.5−45.8i)T2 1 + (5.25 - 9.09i)T + (-26.5 - 45.8i)T^{2}
59 1+(−3.22−1.86i)T+(29.5+51.0i)T2 1 + (-3.22 - 1.86i)T + (29.5 + 51.0i)T^{2}
61 1+(−3.16−5.48i)T+(−30.5+52.8i)T2 1 + (-3.16 - 5.48i)T + (-30.5 + 52.8i)T^{2}
67 1+(10.7+6.19i)T+(33.5+58.0i)T2 1 + (10.7 + 6.19i)T + (33.5 + 58.0i)T^{2}
71 1−6.21iT−71T2 1 - 6.21iT - 71T^{2}
73 1+(8.92+5.15i)T+(36.5+63.2i)T2 1 + (8.92 + 5.15i)T + (36.5 + 63.2i)T^{2}
79 1+(6.41+11.1i)T+(−39.5+68.4i)T2 1 + (6.41 + 11.1i)T + (-39.5 + 68.4i)T^{2}
83 1+5.22iT−83T2 1 + 5.22iT - 83T^{2}
89 1+(−6.94−12.0i)T+(−44.5+77.0i)T2 1 + (-6.94 - 12.0i)T + (-44.5 + 77.0i)T^{2}
97 1−17.1iT−97T2 1 - 17.1iT - 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

−13.31276780185500085030908130235, −12.30122135169566467170152536440, −11.73082946626220198263681322418, −10.26021057681136849594348847931, −8.859090092491976944593210943668, −7.86234427629538080914625420167, −7.06982031578713079564357082532, −6.23235860231301797372988878395, −4.89484806556720266290288042913, −3.05003783288679874829893440389, 0.21611288555846807522497819688, 3.21952884876913770188689905278, 4.34889365175874021826775475584, 4.99545867796691038143973107068, 7.20723516849391404115894213159, 8.483859296318725864888801852752, 9.585564281724804435994711780840, 10.21028578781072586695561495444, 11.39113439213581959178795933644, 12.03384903822452681761342717610

Graph of the ZZ-function along the critical line