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Properties

Label 2-6e4-3.2-c2-0-18
Degree 22
Conductor 12961296
Sign −i-i
Analytic cond. 35.313435.3134
Root an. cond. 5.942515.94251
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  + 5.19i·5-s + 8.34·7-s − 0.953i·11-s − 9.69·13-s + 18.8i·17-s + 24.6·19-s + 0.953i·23-s − 2·25-s + 13.6i·29-s − 3.04·31-s + 43.3i·35-s + 46.6·37-s + 10.9i·41-s − 45.0·43-s − 45.2i·47-s + ⋯
L(s)  = 1  + 1.03i·5-s + 1.19·7-s − 0.0866i·11-s − 0.745·13-s + 1.11i·17-s + 1.29·19-s + 0.0414i·23-s − 0.0800·25-s + 0.471i·29-s − 0.0982·31-s + 1.23i·35-s + 1.26·37-s + 0.266i·41-s − 1.04·43-s − 0.963i·47-s + ⋯

Functional equation

Λ(s)=(1296s/2 ΓC(s) L(s)=(−i Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(1296s/2 ΓC(s+1) L(s)=(−i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12961296    =    24⋅342^{4} \cdot 3^{4}
Sign: −i-i
Analytic conductor: 35.313435.3134
Root analytic conductor: 5.942515.94251
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ1296(161,⋅)\chi_{1296} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1296, ( :1), −i)(2,\ 1296,\ (\ :1),\ -i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.0786302402.078630240
L(12)L(\frac12) ≈\approx 2.0786302402.078630240
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
3 1 1
good5 1−5.19iT−25T2 1 - 5.19iT - 25T^{2}
7 1−8.34T+49T2 1 - 8.34T + 49T^{2}
11 1+0.953iT−121T2 1 + 0.953iT - 121T^{2}
13 1+9.69T+169T2 1 + 9.69T + 169T^{2}
17 1−18.8iT−289T2 1 - 18.8iT - 289T^{2}
19 1−24.6T+361T2 1 - 24.6T + 361T^{2}
23 1−0.953iT−529T2 1 - 0.953iT - 529T^{2}
29 1−13.6iT−841T2 1 - 13.6iT - 841T^{2}
31 1+3.04T+961T2 1 + 3.04T + 961T^{2}
37 1−46.6T+1.36e3T2 1 - 46.6T + 1.36e3T^{2}
41 1−10.9iT−1.68e3T2 1 - 10.9iT - 1.68e3T^{2}
43 1+45.0T+1.84e3T2 1 + 45.0T + 1.84e3T^{2}
47 1+45.2iT−2.20e3T2 1 + 45.2iT - 2.20e3T^{2}
53 1−94.3iT−2.80e3T2 1 - 94.3iT - 2.80e3T^{2}
59 1+18.7iT−3.48e3T2 1 + 18.7iT - 3.48e3T^{2}
61 1−13.0T+3.72e3T2 1 - 13.0T + 3.72e3T^{2}
67 1+75.0T+4.48e3T2 1 + 75.0T + 4.48e3T^{2}
71 1−18.0iT−5.04e3T2 1 - 18.0iT - 5.04e3T^{2}
73 1+7.90T+5.32e3T2 1 + 7.90T + 5.32e3T^{2}
79 1−43.7T+6.24e3T2 1 - 43.7T + 6.24e3T^{2}
83 1−130.iT−6.88e3T2 1 - 130. iT - 6.88e3T^{2}
89 1+145.iT−7.92e3T2 1 + 145. iT - 7.92e3T^{2}
97 1+109.T+9.40e3T2 1 + 109.T + 9.40e3T^{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

−9.817958649565682052190245828312, −8.789811505947656660025757661007, −7.87160936508362327592657273251, −7.35935864449500353002217562485, −6.42479732401661562307382833439, −5.45078866833320293781740234946, −4.60477279413183678912692375328, −3.47557114106265314696308564854, −2.48722870255257886592539478473, −1.33684776117530933015748656369, 0.64340256405058581337037716440, 1.70575107382543915709252610797, 2.94982430615021434405385558993, 4.42843610949763959023458787702, 4.94997701080550235673402033966, 5.60357807915222880898435233247, 6.99312856079008830524280973265, 7.76910173288777908949374559533, 8.365619893364197789143178451909, 9.347132788772198388676182642153

Graph of the ZZ-function along the critical line