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Properties

Label 2-3648-456.365-c0-0-6
Degree 22
Conductor 36483648
Sign 0.796+0.605i0.796 + 0.605i
Analytic cond. 1.820581.82058
Root an. cond. 1.349291.34929
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  + (0.984 + 0.173i)3-s + (0.939 + 0.342i)9-s + (−0.984 − 1.70i)11-s + (0.592 − 1.62i)17-s + (0.642 − 0.766i)19-s + (−0.173 + 0.984i)25-s + (0.866 + 0.5i)27-s + (−0.673 − 1.85i)33-s + (−1.26 + 0.223i)41-s + (−0.642 − 0.766i)43-s + (0.5 + 0.866i)49-s + (0.866 − 1.5i)51-s + (0.766 − 0.642i)57-s + (1.85 + 0.673i)59-s + (−0.524 − 1.43i)67-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)3-s + (0.939 + 0.342i)9-s + (−0.984 − 1.70i)11-s + (0.592 − 1.62i)17-s + (0.642 − 0.766i)19-s + (−0.173 + 0.984i)25-s + (0.866 + 0.5i)27-s + (−0.673 − 1.85i)33-s + (−1.26 + 0.223i)41-s + (−0.642 − 0.766i)43-s + (0.5 + 0.866i)49-s + (0.866 − 1.5i)51-s + (0.766 − 0.642i)57-s + (1.85 + 0.673i)59-s + (−0.524 − 1.43i)67-s + ⋯

Functional equation

Λ(s)=(3648s/2 ΓC(s) L(s)=((0.796+0.605i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3648s/2 ΓC(s) L(s)=((0.796+0.605i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36483648    =    26⋅3⋅192^{6} \cdot 3 \cdot 19
Sign: 0.796+0.605i0.796 + 0.605i
Analytic conductor: 1.820581.82058
Root analytic conductor: 1.349291.34929
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3648(1505,⋅)\chi_{3648} (1505, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3648, ( :0), 0.796+0.605i)(2,\ 3648,\ (\ :0),\ 0.796 + 0.605i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7716882591.771688259
L(12)L(\frac12) ≈\approx 1.7716882591.771688259
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 1
3 1+(−0.984−0.173i)T 1 + (-0.984 - 0.173i)T
19 1+(−0.642+0.766i)T 1 + (-0.642 + 0.766i)T
good5 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
7 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
11 1+(0.984+1.70i)T+(−0.5+0.866i)T2 1 + (0.984 + 1.70i)T + (-0.5 + 0.866i)T^{2}
13 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
17 1+(−0.592+1.62i)T+(−0.766−0.642i)T2 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2}
23 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
29 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(1.26−0.223i)T+(0.939−0.342i)T2 1 + (1.26 - 0.223i)T + (0.939 - 0.342i)T^{2}
43 1+(0.642+0.766i)T+(−0.173+0.984i)T2 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2}
47 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
53 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
59 1+(−1.85−0.673i)T+(0.766+0.642i)T2 1 + (-1.85 - 0.673i)T + (0.766 + 0.642i)T^{2}
61 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
67 1+(0.524+1.43i)T+(−0.766+0.642i)T2 1 + (0.524 + 1.43i)T + (-0.766 + 0.642i)T^{2}
71 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
73 1+(−0.0603−0.342i)T+(−0.939+0.342i)T2 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2}
79 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
83 1+(0.642−1.11i)T+(−0.5−0.866i)T2 1 + (0.642 - 1.11i)T + (-0.5 - 0.866i)T^{2}
89 1+(−1.70−0.300i)T+(0.939+0.342i)T2 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2}
97 1+(−1.43−0.524i)T+(0.766+0.642i)T2 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)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

−8.762010809862934916659358327807, −7.85108638894539889829737042022, −7.45198972553148557193512616093, −6.55510365160858595890345541378, −5.30434768600568420108627641660, −5.08852501163917657357948832792, −3.68279746332861527512513371104, −3.11069637083326590936075746099, −2.45750924604200177437153775365, −0.953782183387318593115696336111, 1.60565034033900899601810210616, 2.23230158869384251179304366468, 3.30157055361918722670584628349, 4.08026591731518938181641787379, 4.87987001330719608855420080461, 5.82324040755104973501310443369, 6.81380425985596889473925895333, 7.43864280397100319588822310277, 8.140805700756887183400755094462, 8.517213896324654279251070930488

Graph of the ZZ-function along the critical line