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Properties

Label 2-1472-1472.1333-c0-0-2
Degree 22
Conductor 14721472
Sign 0.881−0.471i0.881 - 0.471i
Analytic cond. 0.7346230.734623
Root an. cond. 0.8571010.857101
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.608 + 0.793i)2-s + (1.85 − 0.369i)3-s + (−0.258 − 0.965i)4-s + (−0.837 + 1.69i)6-s + (0.923 + 0.382i)8-s + (2.38 − 0.989i)9-s + (−0.837 − 1.69i)12-s + (−0.835 + 1.25i)13-s + (−0.866 + 0.499i)16-s + (−0.669 + 2.49i)18-s + (0.382 + 0.923i)23-s + (1.85 + 0.369i)24-s + (−0.382 + 0.923i)25-s + (−0.483 − 1.42i)26-s + (2.49 − 1.66i)27-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)2-s + (1.85 − 0.369i)3-s + (−0.258 − 0.965i)4-s + (−0.837 + 1.69i)6-s + (0.923 + 0.382i)8-s + (2.38 − 0.989i)9-s + (−0.837 − 1.69i)12-s + (−0.835 + 1.25i)13-s + (−0.866 + 0.499i)16-s + (−0.669 + 2.49i)18-s + (0.382 + 0.923i)23-s + (1.85 + 0.369i)24-s + (−0.382 + 0.923i)25-s + (−0.483 − 1.42i)26-s + (2.49 − 1.66i)27-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=((0.881−0.471i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s) L(s)=((0.881−0.471i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: 0.881−0.471i0.881 - 0.471i
Analytic conductor: 0.7346230.734623
Root analytic conductor: 0.8571010.857101
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1472(1333,⋅)\chi_{1472} (1333, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1472, ( :0), 0.881−0.471i)(2,\ 1472,\ (\ :0),\ 0.881 - 0.471i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4706439101.470643910
L(12)L(\frac12) ≈\approx 1.4706439101.470643910
L(1)L(1) 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+(0.608−0.793i)T 1 + (0.608 - 0.793i)T
23 1+(−0.382−0.923i)T 1 + (-0.382 - 0.923i)T
good3 1+(−1.85+0.369i)T+(0.923−0.382i)T2 1 + (-1.85 + 0.369i)T + (0.923 - 0.382i)T^{2}
5 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
7 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
11 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
13 1+(0.835−1.25i)T+(−0.382−0.923i)T2 1 + (0.835 - 1.25i)T + (-0.382 - 0.923i)T^{2}
17 1+iT2 1 + iT^{2}
19 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
29 1+(0.349+1.75i)T+(−0.923+0.382i)T2 1 + (0.349 + 1.75i)T + (-0.923 + 0.382i)T^{2}
31 1+1.21iT−T2 1 + 1.21iT - T^{2}
37 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
41 1+(0.382+0.923i)T+(−0.707+0.707i)T2 1 + (0.382 + 0.923i)T + (-0.707 + 0.707i)T^{2}
43 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
47 1+(0.366−0.366i)T−iT2 1 + (0.366 - 0.366i)T - iT^{2}
53 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
59 1+(1.08+1.63i)T+(−0.382+0.923i)T2 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2}
61 1+(−0.923+0.382i)T2 1 + (-0.923 + 0.382i)T^{2}
67 1+(−0.923+0.382i)T2 1 + (-0.923 + 0.382i)T^{2}
71 1+(0.241+0.0999i)T+(0.707+0.707i)T2 1 + (0.241 + 0.0999i)T + (0.707 + 0.707i)T^{2}
73 1+(1.78−0.739i)T+(0.707−0.707i)T2 1 + (1.78 - 0.739i)T + (0.707 - 0.707i)T^{2}
79 1−iT2 1 - iT^{2}
83 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
89 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
97 1+T2 1 + 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

−9.546773395626280512113391449201, −8.973522351024259502556943889813, −8.069587177754238668904114290984, −7.48744838397511229860707234320, −7.00511632044461169667171000646, −5.94693816373854407261830375242, −4.59483017604875724778030544075, −3.78915855650649696397254590741, −2.41548845763799947660124788246, −1.63597067266958237522364988625, 1.58692141040650096984411319709, 2.78908988993675154597253490537, 3.10384323064120358332690988212, 4.21329855143058927615634598832, 5.03694543484856123926247836760, 6.95209497977235106579876180787, 7.58780029028992043375632052718, 8.387806354186008501290613260238, 8.780388821236255327996603249342, 9.581395611964288569097073096171

Graph of the ZZ-function along the critical line