Properties

Label 2-1344-4.3-c2-0-47
Degree 22
Conductor 13441344
Sign 1-1
Analytic cond. 36.621336.6213
Root an. cond. 6.051556.05155
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  − 1.73i·3-s + 6.29·5-s + 2.64i·7-s − 2.99·9-s − 16.5i·11-s − 14.1·13-s − 10.8i·15-s − 20.4·17-s − 6.39i·19-s + 4.58·21-s + 29.0i·23-s + 14.5·25-s + 5.19i·27-s − 54.6·29-s − 14.7i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.25·5-s + 0.377i·7-s − 0.333·9-s − 1.50i·11-s − 1.08·13-s − 0.726i·15-s − 1.20·17-s − 0.336i·19-s + 0.218·21-s + 1.26i·23-s + 0.583·25-s + 0.192i·27-s − 1.88·29-s − 0.475i·31-s + ⋯

Functional equation

Λ(s)=(1344s/2ΓC(s)L(s)=(Λ(3s)\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(1344s/2ΓC(s+1)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13441344    =    26372^{6} \cdot 3 \cdot 7
Sign: 1-1
Analytic conductor: 36.621336.6213
Root analytic conductor: 6.051556.05155
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ1344(127,)\chi_{1344} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1344, ( :1), 1)(2,\ 1344,\ (\ :1),\ -1)

Particular Values

L(32)L(\frac{3}{2}) \approx 0.69729801650.6972980165
L(12)L(\frac12) \approx 0.69729801650.6972980165
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+1.73iT 1 + 1.73iT
7 12.64iT 1 - 2.64iT
good5 16.29T+25T2 1 - 6.29T + 25T^{2}
11 1+16.5iT121T2 1 + 16.5iT - 121T^{2}
13 1+14.1T+169T2 1 + 14.1T + 169T^{2}
17 1+20.4T+289T2 1 + 20.4T + 289T^{2}
19 1+6.39iT361T2 1 + 6.39iT - 361T^{2}
23 129.0iT529T2 1 - 29.0iT - 529T^{2}
29 1+54.6T+841T2 1 + 54.6T + 841T^{2}
31 1+14.7iT961T2 1 + 14.7iT - 961T^{2}
37 1+26.8T+1.36e3T2 1 + 26.8T + 1.36e3T^{2}
41 1+33.5T+1.68e3T2 1 + 33.5T + 1.68e3T^{2}
43 1+3.83iT1.84e3T2 1 + 3.83iT - 1.84e3T^{2}
47 1+27.4iT2.20e3T2 1 + 27.4iT - 2.20e3T^{2}
53 121.7T+2.80e3T2 1 - 21.7T + 2.80e3T^{2}
59 154.1iT3.48e3T2 1 - 54.1iT - 3.48e3T^{2}
61 135.7T+3.72e3T2 1 - 35.7T + 3.72e3T^{2}
67 1+95.4iT4.48e3T2 1 + 95.4iT - 4.48e3T^{2}
71 1+55.6iT5.04e3T2 1 + 55.6iT - 5.04e3T^{2}
73 1130.T+5.32e3T2 1 - 130.T + 5.32e3T^{2}
79 115.6iT6.24e3T2 1 - 15.6iT - 6.24e3T^{2}
83 1+49.4iT6.88e3T2 1 + 49.4iT - 6.88e3T^{2}
89 139.2T+7.92e3T2 1 - 39.2T + 7.92e3T^{2}
97 1+171.T+9.40e3T2 1 + 171.T + 9.40e3T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.166990565096219587258027584737, −8.283190687872264232595832780215, −7.29031376756199167462765353891, −6.46871806662027178777662270397, −5.66588382451877625305889519182, −5.18101182989554213760701990290, −3.59592524127049180421950429163, −2.46463181534381209958310861604, −1.74326019648587161615052525785, −0.17113987515956676400818955415, 1.85004898980956121352778875709, 2.47488133900435770534824268457, 3.99764164683947346150407015579, 4.81493957910421931686239927879, 5.48693742897212465494284078196, 6.65683517283155025328272498895, 7.16072853250710587303704395880, 8.360304089005225183439381203463, 9.385883184492363308586961131566, 9.743018613429851354996227892334

Graph of the ZZ-function along the critical line