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Properties

Label 2-21e2-9.2-c2-0-54
Degree 22
Conductor 441441
Sign −0.894−0.447i-0.894 - 0.447i
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  + (−2.37 − 1.37i)2-s + (2.78 − 1.10i)3-s + (1.76 + 3.06i)4-s + (−2.32 + 1.34i)5-s + (−8.14 − 1.18i)6-s + 1.27i·8-s + (6.53 − 6.18i)9-s + 7.37·10-s + (−7.18 − 4.14i)11-s + (8.32 + 6.57i)12-s + (1.91 + 3.31i)13-s + (−4.99 + 6.32i)15-s + (8.82 − 15.2i)16-s − 16.8i·17-s + (−24.0 + 5.73i)18-s − 9.54·19-s + ⋯
L(s)  = 1  + (−1.18 − 0.686i)2-s + (0.929 − 0.369i)3-s + (0.441 + 0.765i)4-s + (−0.465 + 0.268i)5-s + (−1.35 − 0.197i)6-s + 0.159i·8-s + (0.726 − 0.687i)9-s + 0.737·10-s + (−0.653 − 0.377i)11-s + (0.693 + 0.547i)12-s + (0.147 + 0.255i)13-s + (−0.332 + 0.421i)15-s + (0.551 − 0.954i)16-s − 0.993i·17-s + (−1.33 + 0.318i)18-s − 0.502·19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: −0.894−0.447i-0.894 - 0.447i
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), −0.894−0.447i)(2,\ 441,\ (\ :1),\ -0.894 - 0.447i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0849704+0.359732i0.0849704 + 0.359732i
L(12)L(\frac12) ≈\approx 0.0849704+0.359732i0.0849704 + 0.359732i
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+(−2.78+1.10i)T 1 + (-2.78 + 1.10i)T
7 1 1
good2 1+(2.37+1.37i)T+(2+3.46i)T2 1 + (2.37 + 1.37i)T + (2 + 3.46i)T^{2}
5 1+(2.32−1.34i)T+(12.5−21.6i)T2 1 + (2.32 - 1.34i)T + (12.5 - 21.6i)T^{2}
11 1+(7.18+4.14i)T+(60.5+104.i)T2 1 + (7.18 + 4.14i)T + (60.5 + 104. i)T^{2}
13 1+(−1.91−3.31i)T+(−84.5+146.i)T2 1 + (-1.91 - 3.31i)T + (-84.5 + 146. i)T^{2}
17 1+16.8iT−289T2 1 + 16.8iT - 289T^{2}
19 1+9.54T+361T2 1 + 9.54T + 361T^{2}
23 1+(21.1−12.2i)T+(264.5−458.i)T2 1 + (21.1 - 12.2i)T + (264.5 - 458. i)T^{2}
29 1+(41.1+23.7i)T+(420.5+728.i)T2 1 + (41.1 + 23.7i)T + (420.5 + 728. i)T^{2}
31 1+(19.6+33.9i)T+(−480.5+832.i)T2 1 + (19.6 + 33.9i)T + (-480.5 + 832. i)T^{2}
37 1−46.6T+1.36e3T2 1 - 46.6T + 1.36e3T^{2}
41 1+(51.2−29.5i)T+(840.5−1.45e3i)T2 1 + (51.2 - 29.5i)T + (840.5 - 1.45e3i)T^{2}
43 1+(−28.0+48.5i)T+(−924.5−1.60e3i)T2 1 + (-28.0 + 48.5i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(18.6+10.7i)T+(1.10e3+1.91e3i)T2 1 + (18.6 + 10.7i)T + (1.10e3 + 1.91e3i)T^{2}
53 1+25.6iT−2.80e3T2 1 + 25.6iT - 2.80e3T^{2}
59 1+(40.1−23.1i)T+(1.74e3−3.01e3i)T2 1 + (40.1 - 23.1i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(30.1−52.2i)T+(−1.86e3−3.22e3i)T2 1 + (30.1 - 52.2i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−32.0−55.4i)T+(−2.24e3+3.88e3i)T2 1 + (-32.0 - 55.4i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1−49.6iT−5.04e3T2 1 - 49.6iT - 5.04e3T^{2}
73 1+117.T+5.32e3T2 1 + 117.T + 5.32e3T^{2}
79 1+(20.2−35.1i)T+(−3.12e3−5.40e3i)T2 1 + (20.2 - 35.1i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(64.7+37.3i)T+(3.44e3+5.96e3i)T2 1 + (64.7 + 37.3i)T + (3.44e3 + 5.96e3i)T^{2}
89 1+69.2iT−7.92e3T2 1 + 69.2iT - 7.92e3T^{2}
97 1+(46.9−81.3i)T+(−4.70e3−8.14e3i)T2 1 + (46.9 - 81.3i)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

−10.12729361494691039001021904127, −9.493084977232850497874802966824, −8.678810062408868547499346258254, −7.78769242606292771734467728744, −7.33428022338668715620517932434, −5.74550863083521439432024236388, −4.00389518662516753484055277204, −2.83440729951119438354144098383, −1.83104542501761608913831086084, −0.19670121821897849674900999754, 1.81260871833609993223277356948, 3.50307841922487776241627815324, 4.54865550607639373984479139739, 6.07802223319684042175044450169, 7.31494083128697698287602284218, 8.006791559196470819257702216840, 8.537146432192320028632759007378, 9.391571659386476180826888605579, 10.25680192866760298900286715669, 10.86339379074746413613919138882

Graph of the ZZ-function along the critical line