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Properties

Label 2-126-63.2-c2-0-14
Degree 22
Conductor 126126
Sign 0.484+0.874i0.484 + 0.874i
Analytic cond. 3.433253.43325
Root an. cond. 1.852901.85290
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  + (1.22 − 0.707i)2-s + (2.84 − 0.954i)3-s + (0.999 − 1.73i)4-s − 2.12i·5-s + (2.80 − 3.18i)6-s + (−5.95 − 3.68i)7-s − 2.82i·8-s + (7.17 − 5.43i)9-s + (−1.50 − 2.60i)10-s + 8.95i·11-s + (1.19 − 5.88i)12-s + (10.4 + 18.0i)13-s + (−9.89 − 0.305i)14-s + (−2.03 − 6.05i)15-s + (−2.00 − 3.46i)16-s + (−9.01 + 5.20i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.948 − 0.318i)3-s + (0.249 − 0.433i)4-s − 0.425i·5-s + (0.468 − 0.530i)6-s + (−0.850 − 0.526i)7-s − 0.353i·8-s + (0.797 − 0.603i)9-s + (−0.150 − 0.260i)10-s + 0.814i·11-s + (0.0992 − 0.490i)12-s + (0.801 + 1.38i)13-s + (−0.706 − 0.0218i)14-s + (−0.135 − 0.403i)15-s + (−0.125 − 0.216i)16-s + (−0.530 + 0.306i)17-s + ⋯

Functional equation

Λ(s)=(126s/2 ΓC(s) L(s)=((0.484+0.874i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(126s/2 ΓC(s+1) L(s)=((0.484+0.874i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 126126    =    2⋅32⋅72 \cdot 3^{2} \cdot 7
Sign: 0.484+0.874i0.484 + 0.874i
Analytic conductor: 3.433253.43325
Root analytic conductor: 1.852901.85290
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ126(65,⋅)\chi_{126} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 126, ( :1), 0.484+0.874i)(2,\ 126,\ (\ :1),\ 0.484 + 0.874i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.06709−1.21824i2.06709 - 1.21824i
L(12)L(\frac12) ≈\approx 2.06709−1.21824i2.06709 - 1.21824i
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)
bad2 1+(−1.22+0.707i)T 1 + (-1.22 + 0.707i)T
3 1+(−2.84+0.954i)T 1 + (-2.84 + 0.954i)T
7 1+(5.95+3.68i)T 1 + (5.95 + 3.68i)T
good5 1+2.12iT−25T2 1 + 2.12iT - 25T^{2}
11 1−8.95iT−121T2 1 - 8.95iT - 121T^{2}
13 1+(−10.4−18.0i)T+(−84.5+146.i)T2 1 + (-10.4 - 18.0i)T + (-84.5 + 146. i)T^{2}
17 1+(9.01−5.20i)T+(144.5−250.i)T2 1 + (9.01 - 5.20i)T + (144.5 - 250. i)T^{2}
19 1+(−6.15+10.6i)T+(−180.5−312.i)T2 1 + (-6.15 + 10.6i)T + (-180.5 - 312. i)T^{2}
23 1−19.1iT−529T2 1 - 19.1iT - 529T^{2}
29 1+(28.3+16.3i)T+(420.5+728.i)T2 1 + (28.3 + 16.3i)T + (420.5 + 728. i)T^{2}
31 1+(18.9−32.8i)T+(−480.5−832.i)T2 1 + (18.9 - 32.8i)T + (-480.5 - 832. i)T^{2}
37 1+(−16.9+29.3i)T+(−684.5−1.18e3i)T2 1 + (-16.9 + 29.3i)T + (-684.5 - 1.18e3i)T^{2}
41 1+(28.7−16.6i)T+(840.5−1.45e3i)T2 1 + (28.7 - 16.6i)T + (840.5 - 1.45e3i)T^{2}
43 1+(−10.3+17.9i)T+(−924.5−1.60e3i)T2 1 + (-10.3 + 17.9i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(57.3−33.1i)T+(1.10e3−1.91e3i)T2 1 + (57.3 - 33.1i)T + (1.10e3 - 1.91e3i)T^{2}
53 1+(2.72−1.57i)T+(1.40e3−2.43e3i)T2 1 + (2.72 - 1.57i)T + (1.40e3 - 2.43e3i)T^{2}
59 1+(72.9+42.1i)T+(1.74e3+3.01e3i)T2 1 + (72.9 + 42.1i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(−46.0−79.6i)T+(−1.86e3+3.22e3i)T2 1 + (-46.0 - 79.6i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(−33.3+57.8i)T+(−2.24e3−3.88e3i)T2 1 + (-33.3 + 57.8i)T + (-2.24e3 - 3.88e3i)T^{2}
71 1+115.iT−5.04e3T2 1 + 115. iT - 5.04e3T^{2}
73 1+(19.6+33.9i)T+(−2.66e3+4.61e3i)T2 1 + (19.6 + 33.9i)T + (-2.66e3 + 4.61e3i)T^{2}
79 1+(26.2+45.3i)T+(−3.12e3+5.40e3i)T2 1 + (26.2 + 45.3i)T + (-3.12e3 + 5.40e3i)T^{2}
83 1+(−9.25−5.34i)T+(3.44e3+5.96e3i)T2 1 + (-9.25 - 5.34i)T + (3.44e3 + 5.96e3i)T^{2}
89 1+(−61.9−35.7i)T+(3.96e3+6.85e3i)T2 1 + (-61.9 - 35.7i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(−64.5+111.i)T+(−4.70e3−8.14e3i)T2 1 + (-64.5 + 111. i)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

−13.14479996797346347097866883480, −12.28887116359693337829698354052, −11.00819974111341629291378045633, −9.615918676051241730278800243691, −8.989539867153572362143316874757, −7.33707138242060303920635269882, −6.45780582636273019756826088371, −4.51048394633166130167201529779, −3.44707027080103370919121036717, −1.72113777352975284866981362376, 2.81949366285120572609657062872, 3.64423287998879716248703739361, 5.45368593635453056520136525605, 6.63617926945986927405267477547, 7.999596817957746973698600466556, 8.895256232533217564123357309001, 10.16346097888099306125947252475, 11.21986331778890679239891134698, 12.83507771450016681623076840479, 13.26099099365160308180943089906

Graph of the ZZ-function along the critical line