↑

Properties

Label 2-168-168.5-c1-0-14
Degree 22
Conductor 168168
Sign 0.693−0.720i0.693 - 0.720i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.717 + 1.21i)2-s + (1.11 − 1.32i)3-s + (−0.971 + 1.74i)4-s + (0.337 + 0.195i)5-s + (2.41 + 0.401i)6-s + (1.39 + 2.24i)7-s + (−2.82 + 0.0704i)8-s + (−0.529 − 2.95i)9-s + (0.00458 + 0.551i)10-s + (−0.748 − 1.29i)11-s + (1.24 + 3.23i)12-s + 3.28·13-s + (−1.73 + 3.31i)14-s + (0.634 − 0.232i)15-s + (−2.11 − 3.39i)16-s + (−1.68 − 2.91i)17-s + ⋯
L(s)  = 1  + (0.507 + 0.861i)2-s + (0.641 − 0.766i)3-s + (−0.485 + 0.874i)4-s + (0.151 + 0.0872i)5-s + (0.986 + 0.164i)6-s + (0.527 + 0.849i)7-s + (−0.999 + 0.0249i)8-s + (−0.176 − 0.984i)9-s + (0.00144 + 0.174i)10-s + (−0.225 − 0.390i)11-s + (0.358 + 0.933i)12-s + 0.911·13-s + (−0.464 + 0.885i)14-s + (0.163 − 0.0599i)15-s + (−0.528 − 0.848i)16-s + (−0.407 − 0.706i)17-s + ⋯

Functional equation

Λ(s)=(168s/2 ΓC(s) L(s)=((0.693−0.720i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(168s/2 ΓC(s+1/2) L(s)=((0.693−0.720i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 168168    =    23⋅3⋅72^{3} \cdot 3 \cdot 7
Sign: 0.693−0.720i0.693 - 0.720i
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), 0.693−0.720i)(2,\ 168,\ (\ :1/2),\ 0.693 - 0.720i)

Particular Values

L(1)L(1) ≈\approx 1.55418+0.660921i1.55418 + 0.660921i
L(12)L(\frac12) ≈\approx 1.55418+0.660921i1.55418 + 0.660921i
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+(−0.717−1.21i)T 1 + (-0.717 - 1.21i)T
3 1+(−1.11+1.32i)T 1 + (-1.11 + 1.32i)T
7 1+(−1.39−2.24i)T 1 + (-1.39 - 2.24i)T
good5 1+(−0.337−0.195i)T+(2.5+4.33i)T2 1 + (-0.337 - 0.195i)T + (2.5 + 4.33i)T^{2}
11 1+(0.748+1.29i)T+(−5.5+9.52i)T2 1 + (0.748 + 1.29i)T + (-5.5 + 9.52i)T^{2}
13 1−3.28T+13T2 1 - 3.28T + 13T^{2}
17 1+(1.68+2.91i)T+(−8.5+14.7i)T2 1 + (1.68 + 2.91i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.56−4.43i)T+(−9.5−16.4i)T2 1 + (2.56 - 4.43i)T + (-9.5 - 16.4i)T^{2}
23 1+(4.72+2.72i)T+(11.5+19.9i)T2 1 + (4.72 + 2.72i)T + (11.5 + 19.9i)T^{2}
29 1+4.13T+29T2 1 + 4.13T + 29T^{2}
31 1+(3.60−2.07i)T+(15.5−26.8i)T2 1 + (3.60 - 2.07i)T + (15.5 - 26.8i)T^{2}
37 1+(−7.46−4.31i)T+(18.5+32.0i)T2 1 + (-7.46 - 4.31i)T + (18.5 + 32.0i)T^{2}
41 1−11.1T+41T2 1 - 11.1T + 41T^{2}
43 1+4.79iT−43T2 1 + 4.79iT - 43T^{2}
47 1+(2.51−4.34i)T+(−23.5−40.7i)T2 1 + (2.51 - 4.34i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.499+0.864i)T+(−26.5+45.8i)T2 1 + (0.499 + 0.864i)T + (-26.5 + 45.8i)T^{2}
59 1+(−1.36+0.785i)T+(29.5−51.0i)T2 1 + (-1.36 + 0.785i)T + (29.5 - 51.0i)T^{2}
61 1+(−3.40+5.90i)T+(−30.5−52.8i)T2 1 + (-3.40 + 5.90i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.05+1.76i)T+(33.5−58.0i)T2 1 + (-3.05 + 1.76i)T + (33.5 - 58.0i)T^{2}
71 1−14.3iT−71T2 1 - 14.3iT - 71T^{2}
73 1+(−2.76+1.59i)T+(36.5−63.2i)T2 1 + (-2.76 + 1.59i)T + (36.5 - 63.2i)T^{2}
79 1+(0.239−0.414i)T+(−39.5−68.4i)T2 1 + (0.239 - 0.414i)T + (-39.5 - 68.4i)T^{2}
83 1−17.4iT−83T2 1 - 17.4iT - 83T^{2}
89 1+(−2.54+4.41i)T+(−44.5−77.0i)T2 1 + (-2.54 + 4.41i)T + (-44.5 - 77.0i)T^{2}
97 1−9.00iT−97T2 1 - 9.00iT - 97T^{2}
show more
show less
   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

−13.06184779044745487196500673411, −12.30755213175570381997785737226, −11.29284068373055175806778196402, −9.427914396327494359059316020086, −8.411852968014419067716508023808, −7.88756967181435679821969597360, −6.44671677562867397671723077230, −5.71353656411123395638906381090, −3.99175898905261210996230515162, −2.43401401642573580471252798073, 2.01204065205952192747816406954, 3.73653766108167827150187749057, 4.48311518166199324488276551382, 5.83690754241104142686159390298, 7.68779094598316587303878462038, 8.929749721887275897312341524218, 9.799869400436738809930094962868, 10.87267836071858866350301473627, 11.26734029236166023118629178524, 13.05544679212223144781047519927

Graph of the ZZ-function along the critical line