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Properties

Label 2-21e2-9.2-c2-0-34
Degree 22
Conductor 441441
Sign 0.416−0.909i0.416 - 0.909i
Analytic cond. 12.016312.0163
Root an. cond. 3.466463.46646
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.11 + 0.645i)2-s + (2.77 + 1.13i)3-s + (−1.16 − 2.02i)4-s + (−4.18 + 2.41i)5-s + (2.36 + 3.06i)6-s − 8.17i·8-s + (6.40 + 6.32i)9-s − 6.24·10-s + (11.8 + 6.84i)11-s + (−0.937 − 6.93i)12-s + (8.59 + 14.8i)13-s + (−14.3 + 1.94i)15-s + (0.609 − 1.05i)16-s + 8.21i·17-s + (3.08 + 11.2i)18-s + 13.8·19-s + ⋯
L(s)  = 1  + (0.558 + 0.322i)2-s + (0.925 + 0.379i)3-s + (−0.291 − 0.505i)4-s + (−0.837 + 0.483i)5-s + (0.394 + 0.510i)6-s − 1.02i·8-s + (0.711 + 0.702i)9-s − 0.624·10-s + (1.07 + 0.621i)11-s + (−0.0781 − 0.578i)12-s + (0.661 + 1.14i)13-s + (−0.958 + 0.129i)15-s + (0.0381 − 0.0660i)16-s + 0.483i·17-s + (0.171 + 0.622i)18-s + 0.729·19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.416−0.909i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1) L(s)=((0.416−0.909i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.416−0.909i0.416 - 0.909i
Analytic conductor: 12.016312.0163
Root analytic conductor: 3.466463.46646
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ441(344,⋅)\chi_{441} (344, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1), 0.416−0.909i)(2,\ 441,\ (\ :1),\ 0.416 - 0.909i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.27691+1.46158i2.27691 + 1.46158i
L(12)L(\frac12) ≈\approx 2.27691+1.46158i2.27691 + 1.46158i
L(2)L(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)
bad3 1+(−2.77−1.13i)T 1 + (-2.77 - 1.13i)T
7 1 1
good2 1+(−1.11−0.645i)T+(2+3.46i)T2 1 + (-1.11 - 0.645i)T + (2 + 3.46i)T^{2}
5 1+(4.18−2.41i)T+(12.5−21.6i)T2 1 + (4.18 - 2.41i)T + (12.5 - 21.6i)T^{2}
11 1+(−11.8−6.84i)T+(60.5+104.i)T2 1 + (-11.8 - 6.84i)T + (60.5 + 104. i)T^{2}
13 1+(−8.59−14.8i)T+(−84.5+146.i)T2 1 + (-8.59 - 14.8i)T + (-84.5 + 146. i)T^{2}
17 1−8.21iT−289T2 1 - 8.21iT - 289T^{2}
19 1−13.8T+361T2 1 - 13.8T + 361T^{2}
23 1+(−16.0+9.26i)T+(264.5−458.i)T2 1 + (-16.0 + 9.26i)T + (264.5 - 458. i)T^{2}
29 1+(18.1+10.4i)T+(420.5+728.i)T2 1 + (18.1 + 10.4i)T + (420.5 + 728. i)T^{2}
31 1+(−25.7−44.6i)T+(−480.5+832.i)T2 1 + (-25.7 - 44.6i)T + (-480.5 + 832. i)T^{2}
37 1+27.6T+1.36e3T2 1 + 27.6T + 1.36e3T^{2}
41 1+(−18.4+10.6i)T+(840.5−1.45e3i)T2 1 + (-18.4 + 10.6i)T + (840.5 - 1.45e3i)T^{2}
43 1+(−18.1+31.5i)T+(−924.5−1.60e3i)T2 1 + (-18.1 + 31.5i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(51.4+29.6i)T+(1.10e3+1.91e3i)T2 1 + (51.4 + 29.6i)T + (1.10e3 + 1.91e3i)T^{2}
53 1+93.6iT−2.80e3T2 1 + 93.6iT - 2.80e3T^{2}
59 1+(−1.45+0.838i)T+(1.74e3−3.01e3i)T2 1 + (-1.45 + 0.838i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(−32.7+56.6i)T+(−1.86e3−3.22e3i)T2 1 + (-32.7 + 56.6i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−27.6−47.8i)T+(−2.24e3+3.88e3i)T2 1 + (-27.6 - 47.8i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1+14.2iT−5.04e3T2 1 + 14.2iT - 5.04e3T^{2}
73 1+65.3T+5.32e3T2 1 + 65.3T + 5.32e3T^{2}
79 1+(−3.35+5.80i)T+(−3.12e3−5.40e3i)T2 1 + (-3.35 + 5.80i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−11.6−6.69i)T+(3.44e3+5.96e3i)T2 1 + (-11.6 - 6.69i)T + (3.44e3 + 5.96e3i)T^{2}
89 1−15.8iT−7.92e3T2 1 - 15.8iT - 7.92e3T^{2}
97 1+(40.7−70.5i)T+(−4.70e3−8.14e3i)T2 1 + (40.7 - 70.5i)T + (-4.70e3 - 8.14e3i)T^{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

−11.04337160217026028572688260080, −10.02485653322745498307718418225, −9.239142424512684644444953386515, −8.469022188108446515203642806317, −7.12532067715638294998938289581, −6.63416770550470956309166119387, −5.05276886447841510319356197609, −4.05033813496377478861170211813, −3.51593593330666655240321028640, −1.60041723730684402309789664212, 1.00218103876103759331802866858, 2.92387579210259667240894547787, 3.63184715562200674655951385007, 4.50502123983410696740077032038, 5.90414642231805528624359516097, 7.39339140897793279625738065656, 8.061514896317876128642687453027, 8.764197259720797360559245689945, 9.548473838816242504964308958132, 11.15306651279982722501992439309

Graph of the ZZ-function along the critical line