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Properties

Label 2-168-168.5-c1-0-15
Degree 22
Conductor 168168
Sign 0.849+0.528i0.849 + 0.528i
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  + (−1.26 + 0.631i)2-s + (0.528 − 1.64i)3-s + (1.20 − 1.59i)4-s + (2.66 + 1.54i)5-s + (0.372 + 2.42i)6-s + (−1.46 − 2.20i)7-s + (−0.511 + 2.78i)8-s + (−2.44 − 1.74i)9-s + (−4.34 − 0.264i)10-s + (−0.621 − 1.07i)11-s + (−2.00 − 2.82i)12-s + 5.98·13-s + (3.24 + 1.86i)14-s + (3.95 − 3.58i)15-s + (−1.10 − 3.84i)16-s + (−0.595 − 1.03i)17-s + ⋯
L(s)  = 1  + (−0.894 + 0.446i)2-s + (0.305 − 0.952i)3-s + (0.601 − 0.799i)4-s + (1.19 + 0.688i)5-s + (0.152 + 0.988i)6-s + (−0.552 − 0.833i)7-s + (−0.180 + 0.983i)8-s + (−0.813 − 0.581i)9-s + (−1.37 − 0.0835i)10-s + (−0.187 − 0.324i)11-s + (−0.577 − 0.816i)12-s + 1.66·13-s + (0.866 + 0.498i)14-s + (1.02 − 0.926i)15-s + (−0.277 − 0.960i)16-s + (−0.144 − 0.250i)17-s + ⋯

Functional equation

Λ(s)=(168s/2 ΓC(s) L(s)=((0.849+0.528i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(168s/2 ΓC(s+1/2) L(s)=((0.849+0.528i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 168168    =    23⋅3⋅72^{3} \cdot 3 \cdot 7
Sign: 0.849+0.528i0.849 + 0.528i
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), 0.849+0.528i)(2,\ 168,\ (\ :1/2),\ 0.849 + 0.528i)

Particular Values

L(1)L(1) ≈\approx 0.930409−0.265767i0.930409 - 0.265767i
L(12)L(\frac12) ≈\approx 0.930409−0.265767i0.930409 - 0.265767i
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+(1.26−0.631i)T 1 + (1.26 - 0.631i)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

−13.13397363629929166454289109585, −11.36231598464248258509898896724, −10.57726225495767187167270945693, −9.567720499508888225342437770589, −8.581532782999557114144851975032, −7.40939628229458030138601818048, −6.44240825633041821966262100308, −5.91692881480966728745152835451, −3.04595954920602485963164487899, −1.42247552979914511937626957468, 2.06288580055550487146749036414, 3.53779934452240452370399340784, 5.32411500777280023675178438952, 6.45261460706542591088995079548, 8.505969529882563175688937616572, 8.889153954834529624642533208361, 9.775854861911479327766395474476, 10.54258544731950476723070461494, 11.63885088831734133399559720505, 12.94800372766333154175729724913

Graph of the ZZ-function along the critical line