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Properties

Label 2-182-91.10-c2-0-5
Degree 22
Conductor 182182
Sign 0.964+0.265i0.964 + 0.265i
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 − 3.15i·3-s + (0.999 + 1.73i)4-s + (2.18 + 3.78i)5-s + (−2.23 + 3.86i)6-s + (5.93 + 3.71i)7-s − 2.82i·8-s − 0.964·9-s − 6.17i·10-s + 15.8i·11-s + (5.46 − 3.15i)12-s + (−4.43 + 12.2i)13-s + (−4.64 − 8.74i)14-s + (11.9 − 6.89i)15-s + (−2.00 + 3.46i)16-s + (27.1 − 15.6i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s − 1.05i·3-s + (0.249 + 0.433i)4-s + (0.436 + 0.756i)5-s + (−0.372 + 0.644i)6-s + (0.847 + 0.530i)7-s − 0.353i·8-s − 0.107·9-s − 0.617i·10-s + 1.44i·11-s + (0.455 − 0.263i)12-s + (−0.340 + 0.940i)13-s + (−0.331 − 0.624i)14-s + (0.795 − 0.459i)15-s + (−0.125 + 0.216i)16-s + (1.59 − 0.922i)17-s + ⋯

Functional equation

Λ(s)=(182s/2 ΓC(s) L(s)=((0.964+0.265i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(182s/2 ΓC(s+1) L(s)=((0.964+0.265i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 182182    =    2⋅7⋅132 \cdot 7 \cdot 13
Sign: 0.964+0.265i0.964 + 0.265i
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), 0.964+0.265i)(2,\ 182,\ (\ :1),\ 0.964 + 0.265i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.32007−0.178694i1.32007 - 0.178694i
L(12)L(\frac12) ≈\approx 1.32007−0.178694i1.32007 - 0.178694i
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+(−5.93−3.71i)T 1 + (-5.93 - 3.71i)T
13 1+(4.43−12.2i)T 1 + (4.43 - 12.2i)T
good3 1+3.15iT−9T2 1 + 3.15iT - 9T^{2}
5 1+(−2.18−3.78i)T+(−12.5+21.6i)T2 1 + (-2.18 - 3.78i)T + (-12.5 + 21.6i)T^{2}
11 1−15.8iT−121T2 1 - 15.8iT - 121T^{2}
17 1+(−27.1+15.6i)T+(144.5−250.i)T2 1 + (-27.1 + 15.6i)T + (144.5 - 250. i)T^{2}
19 1+14.5T+361T2 1 + 14.5T + 361T^{2}
23 1+(−7.40+12.8i)T+(−264.5−458.i)T2 1 + (-7.40 + 12.8i)T + (-264.5 - 458. i)T^{2}
29 1+(0.175+0.304i)T+(−420.5+728.i)T2 1 + (0.175 + 0.304i)T + (-420.5 + 728. i)T^{2}
31 1+(1.96−3.40i)T+(−480.5−832.i)T2 1 + (1.96 - 3.40i)T + (-480.5 - 832. i)T^{2}
37 1+(4.24+2.44i)T+(684.5+1.18e3i)T2 1 + (4.24 + 2.44i)T + (684.5 + 1.18e3i)T^{2}
41 1+(−33.1−57.4i)T+(−840.5+1.45e3i)T2 1 + (-33.1 - 57.4i)T + (-840.5 + 1.45e3i)T^{2}
43 1+(24.7−42.9i)T+(−924.5−1.60e3i)T2 1 + (24.7 - 42.9i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(42.0+72.7i)T+(−1.10e3+1.91e3i)T2 1 + (42.0 + 72.7i)T + (-1.10e3 + 1.91e3i)T^{2}
53 1+(−9.86+17.0i)T+(−1.40e3−2.43e3i)T2 1 + (-9.86 + 17.0i)T + (-1.40e3 - 2.43e3i)T^{2}
59 1+(34.9+60.6i)T+(−1.74e3+3.01e3i)T2 1 + (34.9 + 60.6i)T + (-1.74e3 + 3.01e3i)T^{2}
61 1+14.7iT−3.72e3T2 1 + 14.7iT - 3.72e3T^{2}
67 1−116.iT−4.48e3T2 1 - 116. iT - 4.48e3T^{2}
71 1+(−9.64−5.56i)T+(2.52e3+4.36e3i)T2 1 + (-9.64 - 5.56i)T + (2.52e3 + 4.36e3i)T^{2}
73 1+(4.36−7.56i)T+(−2.66e3−4.61e3i)T2 1 + (4.36 - 7.56i)T + (-2.66e3 - 4.61e3i)T^{2}
79 1+(55.6+96.3i)T+(−3.12e3+5.40e3i)T2 1 + (55.6 + 96.3i)T + (-3.12e3 + 5.40e3i)T^{2}
83 1−140.T+6.88e3T2 1 - 140.T + 6.88e3T^{2}
89 1+(−57.5+99.7i)T+(−3.96e3−6.85e3i)T2 1 + (-57.5 + 99.7i)T + (-3.96e3 - 6.85e3i)T^{2}
97 1+(5.49−9.52i)T+(−4.70e3−8.14e3i)T2 1 + (5.49 - 9.52i)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

−12.17478381741298765244685280335, −11.59845823946232325583441087965, −10.27198772116579806280368299842, −9.504118347665308260910302870147, −8.114322590568527899201483891498, −7.25145893661840522820179395658, −6.49216862712457557688753734473, −4.74451569852455989381686929327, −2.53527112543706937844817367099, −1.61924998060201331553793882448, 1.15621916337684214638239892362, 3.61062381713325729645647282302, 5.06885745752111661364100330311, 5.80240447781985230719314829290, 7.65349803586409647815279788146, 8.485405925185420424209880705120, 9.421406182786219863039304530442, 10.48978450198810685499606640674, 10.89406603178818525843069129028, 12.40727228689231755334338990303

Graph of the ZZ-function along the critical line