↑

Properties

Label 2-162-81.22-c1-0-3
Degree 22
Conductor 162162
Sign −0.0165−0.999i-0.0165 - 0.999i
Analytic cond. 1.293571.29357
Root an. cond. 1.137351.13735
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  + (0.686 + 0.727i)2-s + (−0.301 + 1.70i)3-s + (−0.0581 + 0.998i)4-s + (1.83 − 0.214i)5-s + (−1.44 + 0.950i)6-s + (−0.0854 − 0.0428i)7-s + (−0.766 + 0.642i)8-s + (−2.81 − 1.03i)9-s + (1.41 + 1.18i)10-s + (0.254 + 0.589i)11-s + (−1.68 − 0.400i)12-s + (−1.34 − 0.319i)13-s + (−0.0274 − 0.0915i)14-s + (−0.188 + 3.19i)15-s + (−0.993 − 0.116i)16-s + (0.845 − 4.79i)17-s + ⋯
L(s)  = 1  + (0.485 + 0.514i)2-s + (−0.174 + 0.984i)3-s + (−0.0290 + 0.499i)4-s + (0.821 − 0.0960i)5-s + (−0.591 + 0.388i)6-s + (−0.0322 − 0.0162i)7-s + (−0.270 + 0.227i)8-s + (−0.939 − 0.343i)9-s + (0.448 + 0.376i)10-s + (0.0767 + 0.177i)11-s + (−0.486 − 0.115i)12-s + (−0.373 − 0.0885i)13-s + (−0.00732 − 0.0244i)14-s + (−0.0486 + 0.826i)15-s + (−0.248 − 0.0290i)16-s + (0.205 − 1.16i)17-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=((−0.0165−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+1/2) L(s)=((−0.0165−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: −0.0165−0.999i-0.0165 - 0.999i
Analytic conductor: 1.293571.29357
Root analytic conductor: 1.137351.13735
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ162(103,⋅)\chi_{162} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 162, ( :1/2), −0.0165−0.999i)(2,\ 162,\ (\ :1/2),\ -0.0165 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.01954+1.03660i1.01954 + 1.03660i
L(12)L(\frac12) ≈\approx 1.01954+1.03660i1.01954 + 1.03660i
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+(−0.686−0.727i)T 1 + (-0.686 - 0.727i)T
3 1+(0.301−1.70i)T 1 + (0.301 - 1.70i)T
good5 1+(−1.83+0.214i)T+(4.86−1.15i)T2 1 + (-1.83 + 0.214i)T + (4.86 - 1.15i)T^{2}
7 1+(0.0854+0.0428i)T+(4.18+5.61i)T2 1 + (0.0854 + 0.0428i)T + (4.18 + 5.61i)T^{2}
11 1+(−0.254−0.589i)T+(−7.54+8.00i)T2 1 + (-0.254 - 0.589i)T + (-7.54 + 8.00i)T^{2}
13 1+(1.34+0.319i)T+(11.6+5.83i)T2 1 + (1.34 + 0.319i)T + (11.6 + 5.83i)T^{2}
17 1+(−0.845+4.79i)T+(−15.9−5.81i)T2 1 + (-0.845 + 4.79i)T + (-15.9 - 5.81i)T^{2}
19 1+(−0.680−3.85i)T+(−17.8+6.49i)T2 1 + (-0.680 - 3.85i)T + (-17.8 + 6.49i)T^{2}
23 1+(−6.15+3.08i)T+(13.7−18.4i)T2 1 + (-6.15 + 3.08i)T + (13.7 - 18.4i)T^{2}
29 1+(−1.82+6.08i)T+(−24.2−15.9i)T2 1 + (-1.82 + 6.08i)T + (-24.2 - 15.9i)T^{2}
31 1+(−0.00237−0.00156i)T+(12.2+28.4i)T2 1 + (-0.00237 - 0.00156i)T + (12.2 + 28.4i)T^{2}
37 1+(−4.08+1.48i)T+(28.3−23.7i)T2 1 + (-4.08 + 1.48i)T + (28.3 - 23.7i)T^{2}
41 1+(−5.48+5.81i)T+(−2.38−40.9i)T2 1 + (-5.48 + 5.81i)T + (-2.38 - 40.9i)T^{2}
43 1+(6.66−8.94i)T+(−12.3−41.1i)T2 1 + (6.66 - 8.94i)T + (-12.3 - 41.1i)T^{2}
47 1+(7.83−5.15i)T+(18.6−43.1i)T2 1 + (7.83 - 5.15i)T + (18.6 - 43.1i)T^{2}
53 1+(5.22−9.04i)T+(−26.5−45.8i)T2 1 + (5.22 - 9.04i)T + (-26.5 - 45.8i)T^{2}
59 1+(−1.22+2.83i)T+(−40.4−42.9i)T2 1 + (-1.22 + 2.83i)T + (-40.4 - 42.9i)T^{2}
61 1+(−0.152−2.61i)T+(−60.5+7.08i)T2 1 + (-0.152 - 2.61i)T + (-60.5 + 7.08i)T^{2}
67 1+(2.24+7.48i)T+(−55.9+36.8i)T2 1 + (2.24 + 7.48i)T + (-55.9 + 36.8i)T^{2}
71 1+(6.01+5.04i)T+(12.3+69.9i)T2 1 + (6.01 + 5.04i)T + (12.3 + 69.9i)T^{2}
73 1+(5.97−5.01i)T+(12.6−71.8i)T2 1 + (5.97 - 5.01i)T + (12.6 - 71.8i)T^{2}
79 1+(−7.38−7.82i)T+(−4.59+78.8i)T2 1 + (-7.38 - 7.82i)T + (-4.59 + 78.8i)T^{2}
83 1+(1.46+1.54i)T+(−4.82+82.8i)T2 1 + (1.46 + 1.54i)T + (-4.82 + 82.8i)T^{2}
89 1+(3.99−3.35i)T+(15.4−87.6i)T2 1 + (3.99 - 3.35i)T + (15.4 - 87.6i)T^{2}
97 1+(16.4+1.91i)T+(94.3+22.3i)T2 1 + (16.4 + 1.91i)T + (94.3 + 22.3i)T^{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

−13.31429054763881762343011959671, −12.16880077531046549933269361191, −11.13996067568577300437728914169, −9.879874264117062905721515552043, −9.293945443727560315434749202955, −7.908143496750262631010723413374, −6.43617979090827547523258158762, −5.42489671824064235418457653745, −4.48110640114763917974796793336, −2.90096710895502153972384092494, 1.61203687055671515347359088388, 3.04043683460499184371714797251, 5.05893971616537352572284284806, 6.10318397737196861875836239841, 7.07471236372562025794388005927, 8.537194176915103782346453718978, 9.716007120414654376050608146309, 10.87799831661095450220578360950, 11.71378941014149464649669560038, 12.86941843897302290293406185070

Graph of the ZZ-function along the critical line