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Properties

Label 2-3e5-9.4-c1-0-7
Degree 22
Conductor 243243
Sign 0.766−0.642i0.766 - 0.642i
Analytic cond. 1.940361.94036
Root an. cond. 1.392961.39296
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  + (0.866 + 1.5i)2-s + (−0.5 + 0.866i)4-s + (1.73 − 3i)5-s + (0.5 + 0.866i)7-s + 1.73·8-s + 6·10-s + (−1.73 − 3i)11-s + (−2.5 + 4.33i)13-s + (−0.866 + 1.5i)14-s + (2.49 + 4.33i)16-s − 19-s + (1.73 + 3i)20-s + (3 − 5.19i)22-s + (−3.46 + 6i)23-s + (−3.5 − 6.06i)25-s − 8.66·26-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (−0.250 + 0.433i)4-s + (0.774 − 1.34i)5-s + (0.188 + 0.327i)7-s + 0.612·8-s + 1.89·10-s + (−0.522 − 0.904i)11-s + (−0.693 + 1.20i)13-s + (−0.231 + 0.400i)14-s + (0.624 + 1.08i)16-s − 0.229·19-s + (0.387 + 0.670i)20-s + (0.639 − 1.10i)22-s + (−0.722 + 1.25i)23-s + (−0.700 − 1.21i)25-s − 1.69·26-s + ⋯

Functional equation

Λ(s)=(243s/2 ΓC(s) L(s)=((0.766−0.642i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(243s/2 ΓC(s+1/2) L(s)=((0.766−0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 243243    =    353^{5}
Sign: 0.766−0.642i0.766 - 0.642i
Analytic conductor: 1.940361.94036
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ243(82,⋅)\chi_{243} (82, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 243, ( :1/2), 0.766−0.642i)(2,\ 243,\ (\ :1/2),\ 0.766 - 0.642i)

Particular Values

L(1)L(1) ≈\approx 1.79165+0.652107i1.79165 + 0.652107i
L(12)L(\frac12) ≈\approx 1.79165+0.652107i1.79165 + 0.652107i
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)
bad3 1 1
good2 1+(−0.866−1.5i)T+(−1+1.73i)T2 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2}
5 1+(−1.73+3i)T+(−2.5−4.33i)T2 1 + (-1.73 + 3i)T + (-2.5 - 4.33i)T^{2}
7 1+(−0.5−0.866i)T+(−3.5+6.06i)T2 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.73+3i)T+(−5.5+9.52i)T2 1 + (1.73 + 3i)T + (-5.5 + 9.52i)T^{2}
13 1+(2.5−4.33i)T+(−6.5−11.2i)T2 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2}
17 1+17T2 1 + 17T^{2}
19 1+T+19T2 1 + T + 19T^{2}
23 1+(3.46−6i)T+(−11.5−19.9i)T2 1 + (3.46 - 6i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.73+3i)T+(−14.5+25.1i)T2 1 + (1.73 + 3i)T + (-14.5 + 25.1i)T^{2}
31 1+(2.5−4.33i)T+(−15.5−26.8i)T2 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1+(−1.73+3i)T+(−20.5−35.5i)T2 1 + (-1.73 + 3i)T + (-20.5 - 35.5i)T^{2}
43 1+(−0.5−0.866i)T+(−21.5+37.2i)T2 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2}
47 1+(1.73+3i)T+(−23.5+40.7i)T2 1 + (1.73 + 3i)T + (-23.5 + 40.7i)T^{2}
53 1−10.3T+53T2 1 - 10.3T + 53T^{2}
59 1+(−1.73+3i)T+(−29.5−51.0i)T2 1 + (-1.73 + 3i)T + (-29.5 - 51.0i)T^{2}
61 1+(1+1.73i)T+(−30.5+52.8i)T2 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2}
67 1+(4−6.92i)T+(−33.5−58.0i)T2 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2}
71 1+10.3T+71T2 1 + 10.3T + 71T^{2}
73 1−2T+73T2 1 - 2T + 73T^{2}
79 1+(−0.5−0.866i)T+(−39.5+68.4i)T2 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2}
83 1+(−3.46−6i)T+(−41.5+71.8i)T2 1 + (-3.46 - 6i)T + (-41.5 + 71.8i)T^{2}
89 1+10.3T+89T2 1 + 10.3T + 89T^{2}
97 1+(8.5+14.7i)T+(−48.5+84.0i)T2 1 + (8.5 + 14.7i)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

−12.50514349076677801374128331738, −11.48282574442425020667876943096, −10.09868768383671068710163975115, −9.070009141545368114370431480391, −8.241147477535537178114824731863, −7.05956508171912337153803989178, −5.76198737946871136007362888362, −5.32027457550671205414922003503, −4.20465437540909186227380024054, −1.83041881739976419156957473316, 2.17825215517418139029996194373, 2.94595583608623063452189652700, 4.37094827743046545745590114849, 5.64648580688680913343882915749, 7.02881533533033942315144179759, 7.82704421819917689819318639729, 9.735941098852421848983030941166, 10.48038938322974848562198398821, 10.78640462950096596729410111834, 12.08927992941335578009892212926

Graph of the ZZ-function along the critical line