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Properties

Label 2-1472-1472.45-c0-0-2
Degree 22
Conductor 14721472
Sign 0.956−0.290i0.956 - 0.290i
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  + (0.793 − 0.608i)2-s + (0.996 + 1.49i)3-s + (0.258 − 0.965i)4-s + (1.69 + 0.576i)6-s + (−0.382 − 0.923i)8-s + (−0.848 + 2.04i)9-s + (1.69 − 0.576i)12-s + (−0.0255 + 0.128i)13-s + (−0.866 − 0.499i)16-s + (0.573 + 2.14i)18-s + (0.923 + 0.382i)23-s + (0.996 − 1.49i)24-s + (−0.923 + 0.382i)25-s + (0.0578 + 0.117i)26-s + (−2.14 + 0.426i)27-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (0.996 + 1.49i)3-s + (0.258 − 0.965i)4-s + (1.69 + 0.576i)6-s + (−0.382 − 0.923i)8-s + (−0.848 + 2.04i)9-s + (1.69 − 0.576i)12-s + (−0.0255 + 0.128i)13-s + (−0.866 − 0.499i)16-s + (0.573 + 2.14i)18-s + (0.923 + 0.382i)23-s + (0.996 − 1.49i)24-s + (−0.923 + 0.382i)25-s + (0.0578 + 0.117i)26-s + (−2.14 + 0.426i)27-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=((0.956−0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s) L(s)=((0.956−0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: 0.956−0.290i0.956 - 0.290i
Analytic conductor: 0.7346230.734623
Root analytic conductor: 0.8571010.857101
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1472(45,⋅)\chi_{1472} (45, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1472, ( :0), 0.956−0.290i)(2,\ 1472,\ (\ :0),\ 0.956 - 0.290i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.2122190742.212219074
L(12)L(\frac12) ≈\approx 2.2122190742.212219074
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+(−0.793+0.608i)T 1 + (-0.793 + 0.608i)T
23 1+(−0.923−0.382i)T 1 + (-0.923 - 0.382i)T
good3 1+(−0.996−1.49i)T+(−0.382+0.923i)T2 1 + (-0.996 - 1.49i)T + (-0.382 + 0.923i)T^{2}
5 1+(0.923−0.382i)T2 1 + (0.923 - 0.382i)T^{2}
7 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
11 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
13 1+(0.0255−0.128i)T+(−0.923−0.382i)T2 1 + (0.0255 - 0.128i)T + (-0.923 - 0.382i)T^{2}
17 1−iT2 1 - iT^{2}
19 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
29 1+(−0.534+0.357i)T+(0.382−0.923i)T2 1 + (-0.534 + 0.357i)T + (0.382 - 0.923i)T^{2}
31 1+1.58iT−T2 1 + 1.58iT - T^{2}
37 1+(−0.923+0.382i)T2 1 + (-0.923 + 0.382i)T^{2}
41 1+(0.923+0.382i)T+(0.707+0.707i)T2 1 + (0.923 + 0.382i)T + (0.707 + 0.707i)T^{2}
43 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
47 1+(0.366+0.366i)T+iT2 1 + (0.366 + 0.366i)T + iT^{2}
53 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
59 1+(0.216+1.08i)T+(−0.923+0.382i)T2 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2}
61 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
67 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
71 1+(0.758+1.83i)T+(−0.707+0.707i)T2 1 + (0.758 + 1.83i)T + (-0.707 + 0.707i)T^{2}
73 1+(0.739−1.78i)T+(−0.707−0.707i)T2 1 + (0.739 - 1.78i)T + (-0.707 - 0.707i)T^{2}
79 1+iT2 1 + iT^{2}
83 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)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

−9.729888826586958383556505285740, −9.321923383710288727650359690149, −8.400564493487693475555958141878, −7.42269615365605785112358305589, −6.13337314763184219359456288258, −5.18944794286851368948634978607, −4.49184378923829077023423206270, −3.71207868526650239601602077435, −3.00847442449069907432304846725, −2.00174920391195998122248364421, 1.61210529516156052320820393839, 2.76576470196142004438640951467, 3.38715310997106749300321994899, 4.66126978984204951093912704311, 5.76962261611922621203912069702, 6.69260700264179948398905855023, 7.06936667805800211352064037648, 7.976924962315707449962961760406, 8.486196515211007956700834465562, 9.182391304213612287744350284788

Graph of the ZZ-function along the critical line