↑

Properties

Label 2-161-161.45-c1-0-9
Degree 22
Conductor 161161
Sign 0.932+0.361i0.932 + 0.361i
Analytic cond. 1.285591.28559
Root an. cond. 1.133831.13383
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  + (0.292 + 0.506i)2-s + (0.510 + 0.294i)3-s + (0.829 − 1.43i)4-s + (−1.76 − 3.05i)5-s + 0.344i·6-s + (2.22 + 1.43i)7-s + 2.13·8-s + (−1.32 − 2.29i)9-s + (1.03 − 1.78i)10-s + (−0.243 − 0.140i)11-s + (0.846 − 0.488i)12-s + 6.20i·13-s + (−0.0792 + 1.54i)14-s − 2.08i·15-s + (−1.03 − 1.79i)16-s + (−1.89 + 3.28i)17-s + ⋯
L(s)  = 1  + (0.206 + 0.357i)2-s + (0.294 + 0.170i)3-s + (0.414 − 0.718i)4-s + (−0.790 − 1.36i)5-s + 0.140i·6-s + (0.839 + 0.543i)7-s + 0.755·8-s + (−0.442 − 0.765i)9-s + (0.326 − 0.565i)10-s + (−0.0733 − 0.0423i)11-s + (0.244 − 0.141i)12-s + 1.72i·13-s + (−0.0211 + 0.412i)14-s − 0.537i·15-s + (−0.258 − 0.447i)16-s + (−0.460 + 0.797i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((0.932+0.361i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+1/2) L(s)=((0.932+0.361i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: 0.932+0.361i0.932 + 0.361i
Analytic conductor: 1.285591.28559
Root analytic conductor: 1.133831.13383
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ161(45,⋅)\chi_{161} (45, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :1/2), 0.932+0.361i)(2,\ 161,\ (\ :1/2),\ 0.932 + 0.361i)

Particular Values

L(1)L(1) ≈\approx 1.37428−0.257465i1.37428 - 0.257465i
L(12)L(\frac12) ≈\approx 1.37428−0.257465i1.37428 - 0.257465i
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)
bad7 1+(−2.22−1.43i)T 1 + (-2.22 - 1.43i)T
23 1+(−4.78+0.372i)T 1 + (-4.78 + 0.372i)T
good2 1+(−0.292−0.506i)T+(−1+1.73i)T2 1 + (-0.292 - 0.506i)T + (-1 + 1.73i)T^{2}
3 1+(−0.510−0.294i)T+(1.5+2.59i)T2 1 + (-0.510 - 0.294i)T + (1.5 + 2.59i)T^{2}
5 1+(1.76+3.05i)T+(−2.5+4.33i)T2 1 + (1.76 + 3.05i)T + (-2.5 + 4.33i)T^{2}
11 1+(0.243+0.140i)T+(5.5+9.52i)T2 1 + (0.243 + 0.140i)T + (5.5 + 9.52i)T^{2}
13 1−6.20iT−13T2 1 - 6.20iT - 13T^{2}
17 1+(1.89−3.28i)T+(−8.5−14.7i)T2 1 + (1.89 - 3.28i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.92−3.33i)T+(−9.5+16.4i)T2 1 + (-1.92 - 3.33i)T + (-9.5 + 16.4i)T^{2}
29 1+2.64T+29T2 1 + 2.64T + 29T^{2}
31 1+(4.59+2.65i)T+(15.5+26.8i)T2 1 + (4.59 + 2.65i)T + (15.5 + 26.8i)T^{2}
37 1+(−0.645+0.372i)T+(18.5−32.0i)T2 1 + (-0.645 + 0.372i)T + (18.5 - 32.0i)T^{2}
41 1+4.20iT−41T2 1 + 4.20iT - 41T^{2}
43 1−7.90iT−43T2 1 - 7.90iT - 43T^{2}
47 1+(5.41−3.12i)T+(23.5−40.7i)T2 1 + (5.41 - 3.12i)T + (23.5 - 40.7i)T^{2}
53 1+(−1.15−0.664i)T+(26.5+45.8i)T2 1 + (-1.15 - 0.664i)T + (26.5 + 45.8i)T^{2}
59 1+(−3.11−1.79i)T+(29.5+51.0i)T2 1 + (-3.11 - 1.79i)T + (29.5 + 51.0i)T^{2}
61 1+(5.17+8.96i)T+(−30.5+52.8i)T2 1 + (5.17 + 8.96i)T + (-30.5 + 52.8i)T^{2}
67 1+(9.69+5.59i)T+(33.5+58.0i)T2 1 + (9.69 + 5.59i)T + (33.5 + 58.0i)T^{2}
71 1−5.77T+71T2 1 - 5.77T + 71T^{2}
73 1+(−6.45−3.72i)T+(36.5+63.2i)T2 1 + (-6.45 - 3.72i)T + (36.5 + 63.2i)T^{2}
79 1+(−6.19+3.57i)T+(39.5−68.4i)T2 1 + (-6.19 + 3.57i)T + (39.5 - 68.4i)T^{2}
83 1+10.3T+83T2 1 + 10.3T + 83T^{2}
89 1+(2.79+4.83i)T+(−44.5+77.0i)T2 1 + (2.79 + 4.83i)T + (-44.5 + 77.0i)T^{2}
97 1−10.9T+97T2 1 - 10.9T + 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

−12.73521310062000690000805037602, −11.73193389503808730690571721246, −11.16824333521101360890288687304, −9.397697759433678869627377289637, −8.791245195421682891327452722916, −7.70358020967331542885638566115, −6.25330695811111162206507541230, −5.06751426054817791512220002540, −4.11115985068732770952850420502, −1.63170443547962894657868202379, 2.61020537051098237097086166868, 3.43478732647794967673011897909, 5.04192196988942998692936817554, 7.16508222285407294352513431529, 7.49858012139684080445381488060, 8.445876357314696281722681890122, 10.51633810724519342367937496235, 11.03410119640308966418557875624, 11.63260226006320875218776211353, 13.02594940495310867136902761507

Graph of the ZZ-function along the critical line