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Properties

Label 2-182-91.81-c1-0-2
Degree 22
Conductor 182182
Sign 0.976−0.213i0.976 - 0.213i
Analytic cond. 1.453271.45327
Root an. cond. 1.205511.20551
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s − 2.39·3-s + (−0.499 − 0.866i)4-s + (−0.561 − 0.972i)5-s + (1.19 − 2.07i)6-s + (2.64 − 0.173i)7-s + 0.999·8-s + 2.75·9-s + 1.12·10-s + 5.57·11-s + (1.19 + 2.07i)12-s + (−0.197 + 3.60i)13-s + (−1.16 + 2.37i)14-s + (1.34 + 2.33i)15-s + (−0.5 + 0.866i)16-s + (−3.66 − 6.35i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s − 1.38·3-s + (−0.249 − 0.433i)4-s + (−0.251 − 0.434i)5-s + (0.489 − 0.848i)6-s + (0.997 − 0.0656i)7-s + 0.353·8-s + 0.919·9-s + 0.355·10-s + 1.68·11-s + (0.346 + 0.599i)12-s + (−0.0548 + 0.998i)13-s + (−0.312 + 0.634i)14-s + (0.347 + 0.602i)15-s + (−0.125 + 0.216i)16-s + (−0.889 − 1.54i)17-s + ⋯

Functional equation

Λ(s)=(182s/2 ΓC(s) L(s)=((0.976−0.213i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(182s/2 ΓC(s+1/2) L(s)=((0.976−0.213i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 182182    =    2⋅7⋅132 \cdot 7 \cdot 13
Sign: 0.976−0.213i0.976 - 0.213i
Analytic conductor: 1.453271.45327
Root analytic conductor: 1.205511.20551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ182(81,⋅)\chi_{182} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 182, ( :1/2), 0.976−0.213i)(2,\ 182,\ (\ :1/2),\ 0.976 - 0.213i)

Particular Values

L(1)L(1) ≈\approx 0.692387+0.0749269i0.692387 + 0.0749269i
L(12)L(\frac12) ≈\approx 0.692387+0.0749269i0.692387 + 0.0749269i
L(32)L(\frac{3}{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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
7 1+(−2.64+0.173i)T 1 + (-2.64 + 0.173i)T
13 1+(0.197−3.60i)T 1 + (0.197 - 3.60i)T
good3 1+2.39T+3T2 1 + 2.39T + 3T^{2}
5 1+(0.561+0.972i)T+(−2.5+4.33i)T2 1 + (0.561 + 0.972i)T + (-2.5 + 4.33i)T^{2}
11 1−5.57T+11T2 1 - 5.57T + 11T^{2}
17 1+(3.66+6.35i)T+(−8.5+14.7i)T2 1 + (3.66 + 6.35i)T + (-8.5 + 14.7i)T^{2}
19 1−6.63T+19T2 1 - 6.63T + 19T^{2}
23 1+(−1.69+2.94i)T+(−11.5−19.9i)T2 1 + (-1.69 + 2.94i)T + (-11.5 - 19.9i)T^{2}
29 1+(−0.929−1.60i)T+(−14.5+25.1i)T2 1 + (-0.929 - 1.60i)T + (-14.5 + 25.1i)T^{2}
31 1+(2.60−4.51i)T+(−15.5−26.8i)T2 1 + (2.60 - 4.51i)T + (-15.5 - 26.8i)T^{2}
37 1+(−1.89+3.28i)T+(−18.5−32.0i)T2 1 + (-1.89 + 3.28i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.55+2.69i)T+(−20.5+35.5i)T2 1 + (1.55 + 2.69i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.97−6.89i)T+(−21.5−37.2i)T2 1 + (3.97 - 6.89i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.44+2.51i)T+(−23.5+40.7i)T2 1 + (1.44 + 2.51i)T + (-23.5 + 40.7i)T^{2}
53 1+(5.29−9.16i)T+(−26.5−45.8i)T2 1 + (5.29 - 9.16i)T + (-26.5 - 45.8i)T^{2}
59 1+(2.35+4.07i)T+(−29.5+51.0i)T2 1 + (2.35 + 4.07i)T + (-29.5 + 51.0i)T^{2}
61 1−4.95T+61T2 1 - 4.95T + 61T^{2}
67 1−0.510T+67T2 1 - 0.510T + 67T^{2}
71 1+(−3.07+5.32i)T+(−35.5−61.4i)T2 1 + (-3.07 + 5.32i)T + (-35.5 - 61.4i)T^{2}
73 1+(3.68−6.38i)T+(−36.5−63.2i)T2 1 + (3.68 - 6.38i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.36−4.09i)T+(−39.5+68.4i)T2 1 + (-2.36 - 4.09i)T + (-39.5 + 68.4i)T^{2}
83 1−1.80T+83T2 1 - 1.80T + 83T^{2}
89 1+(4.18−7.24i)T+(−44.5−77.0i)T2 1 + (4.18 - 7.24i)T + (-44.5 - 77.0i)T^{2}
97 1+(−2.08+3.60i)T+(−48.5−84.0i)T2 1 + (-2.08 + 3.60i)T + (-48.5 - 84.0i)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

−12.22688977885848807429170420602, −11.60525968324251502444478339227, −11.02873168776849178144219108776, −9.491860144173119783100285505640, −8.726001532562200799147561993942, −7.18780278202946094143957530049, −6.51701781810564834467559484251, −5.11788121434211840351206013760, −4.48020009047940889844802867918, −1.11258520739586448069622318586, 1.34542170971051231906626574394, 3.68479750323322531421322466146, 5.03802446597916962190101032192, 6.22138867202972524136349153542, 7.42008780356272839301551815768, 8.657888022630951471925542120845, 9.938677299976271073313177470231, 11.04632468529545967877909747830, 11.41045830884318948603543877770, 12.11786121537305731867597581950

Graph of the ZZ-function along the critical line