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Properties

Label 2-182-91.10-c2-0-8
Degree 22
Conductor 182182
Sign −0.642−0.766i-0.642 - 0.766i
Analytic cond. 4.959144.95914
Root an. cond. 2.226912.22691
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 + 4.02i·3-s + (0.999 + 1.73i)4-s + (4.59 + 7.96i)5-s + (−2.84 + 4.93i)6-s + (−5.39 − 4.45i)7-s + 2.82i·8-s − 7.23·9-s + 13.0i·10-s − 17.1i·11-s + (−6.97 + 4.02i)12-s + (9.51 − 8.85i)13-s + (−3.46 − 9.27i)14-s + (−32.0 + 18.5i)15-s + (−2.00 + 3.46i)16-s + (3.18 − 1.83i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + 1.34i·3-s + (0.249 + 0.433i)4-s + (0.919 + 1.59i)5-s + (−0.474 + 0.822i)6-s + (−0.771 − 0.636i)7-s + 0.353i·8-s − 0.803·9-s + 1.30i·10-s − 1.56i·11-s + (−0.581 + 0.335i)12-s + (0.731 − 0.681i)13-s + (−0.247 − 0.662i)14-s + (−2.13 + 1.23i)15-s + (−0.125 + 0.216i)16-s + (0.187 − 0.108i)17-s + ⋯

Functional equation

Λ(s)=(182s/2 ΓC(s) L(s)=((−0.642−0.766i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(182s/2 ΓC(s+1) L(s)=((−0.642−0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 182182    =    2⋅7⋅132 \cdot 7 \cdot 13
Sign: −0.642−0.766i-0.642 - 0.766i
Analytic conductor: 4.959144.95914
Root analytic conductor: 2.226912.22691
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ182(101,⋅)\chi_{182} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 182, ( :1), −0.642−0.766i)(2,\ 182,\ (\ :1),\ -0.642 - 0.766i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.955199+2.04612i0.955199 + 2.04612i
L(12)L(\frac12) ≈\approx 0.955199+2.04612i0.955199 + 2.04612i
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
7 1+(5.39+4.45i)T 1 + (5.39 + 4.45i)T
13 1+(−9.51+8.85i)T 1 + (-9.51 + 8.85i)T
good3 1−4.02iT−9T2 1 - 4.02iT - 9T^{2}
5 1+(−4.59−7.96i)T+(−12.5+21.6i)T2 1 + (-4.59 - 7.96i)T + (-12.5 + 21.6i)T^{2}
11 1+17.1iT−121T2 1 + 17.1iT - 121T^{2}
17 1+(−3.18+1.83i)T+(144.5−250.i)T2 1 + (-3.18 + 1.83i)T + (144.5 - 250. i)T^{2}
19 1−17.2T+361T2 1 - 17.2T + 361T^{2}
23 1+(3.50−6.06i)T+(−264.5−458.i)T2 1 + (3.50 - 6.06i)T + (-264.5 - 458. i)T^{2}
29 1+(−0.359−0.623i)T+(−420.5+728.i)T2 1 + (-0.359 - 0.623i)T + (-420.5 + 728. i)T^{2}
31 1+(22.2−38.4i)T+(−480.5−832.i)T2 1 + (22.2 - 38.4i)T + (-480.5 - 832. i)T^{2}
37 1+(−26.4−15.2i)T+(684.5+1.18e3i)T2 1 + (-26.4 - 15.2i)T + (684.5 + 1.18e3i)T^{2}
41 1+(16.2+28.1i)T+(−840.5+1.45e3i)T2 1 + (16.2 + 28.1i)T + (-840.5 + 1.45e3i)T^{2}
43 1+(−26.8+46.4i)T+(−924.5−1.60e3i)T2 1 + (-26.8 + 46.4i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(36.6+63.5i)T+(−1.10e3+1.91e3i)T2 1 + (36.6 + 63.5i)T + (-1.10e3 + 1.91e3i)T^{2}
53 1+(4.39−7.62i)T+(−1.40e3−2.43e3i)T2 1 + (4.39 - 7.62i)T + (-1.40e3 - 2.43e3i)T^{2}
59 1+(6.58+11.4i)T+(−1.74e3+3.01e3i)T2 1 + (6.58 + 11.4i)T + (-1.74e3 + 3.01e3i)T^{2}
61 1−11.6iT−3.72e3T2 1 - 11.6iT - 3.72e3T^{2}
67 1+27.7iT−4.48e3T2 1 + 27.7iT - 4.48e3T^{2}
71 1+(25.6+14.7i)T+(2.52e3+4.36e3i)T2 1 + (25.6 + 14.7i)T + (2.52e3 + 4.36e3i)T^{2}
73 1+(30.1−52.1i)T+(−2.66e3−4.61e3i)T2 1 + (30.1 - 52.1i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(66.3+114.i)T+(−3.12e3+5.40e3i)T2 1 + (66.3 + 114. i)T + (-3.12e3 + 5.40e3i)T^{2}
83 1−108.T+6.88e3T2 1 - 108.T + 6.88e3T^{2}
89 1+(−48.6+84.2i)T+(−3.96e3−6.85e3i)T2 1 + (-48.6 + 84.2i)T + (-3.96e3 - 6.85e3i)T^{2}
97 1+(−39.5+68.5i)T+(−4.70e3−8.14e3i)T2 1 + (-39.5 + 68.5i)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

−13.26306514178071269586454063631, −11.40715161821027659722073022557, −10.58904183294868373898242444973, −10.14952230836060954263207581163, −8.957216768774234566870326368862, −7.30054426167968013568624582682, −6.19096050940188204403581575337, −5.44831016058329197444665695078, −3.42683101693864448644433377100, −3.30399843619337399645263370641, 1.29914256771974732590517414297, 2.25340215411286514550859949257, 4.46823823905437262286308240755, 5.71830025777063944535852063185, 6.50278615575196987433036858277, 7.84487656937696425651513819008, 9.260487470417592573495067082334, 9.744868041222793161386386317805, 11.68897691165237307380963079792, 12.43846026043508612635075147635

Graph of the ZZ-function along the critical line