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Properties

Label 2-3648-456.149-c0-0-4
Degree 22
Conductor 36483648
Sign 0.994−0.102i0.994 - 0.102i
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  + (−0.342 + 0.939i)3-s + (−0.766 − 0.642i)9-s + (0.342 − 0.592i)11-s + (1.11 − 1.32i)17-s + (−0.984 + 0.173i)19-s + (0.939 + 0.342i)25-s + (0.866 − 0.500i)27-s + (0.439 + 0.524i)33-s + (−0.673 − 1.85i)41-s + (0.984 + 0.173i)43-s + (0.5 − 0.866i)49-s + (0.866 + 1.5i)51-s + (0.173 − 0.984i)57-s + (0.524 + 0.439i)59-s + (−0.223 − 0.266i)67-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)3-s + (−0.766 − 0.642i)9-s + (0.342 − 0.592i)11-s + (1.11 − 1.32i)17-s + (−0.984 + 0.173i)19-s + (0.939 + 0.342i)25-s + (0.866 − 0.500i)27-s + (0.439 + 0.524i)33-s + (−0.673 − 1.85i)41-s + (0.984 + 0.173i)43-s + (0.5 − 0.866i)49-s + (0.866 + 1.5i)51-s + (0.173 − 0.984i)57-s + (0.524 + 0.439i)59-s + (−0.223 − 0.266i)67-s + ⋯

Functional equation

Λ(s)=(3648s/2 ΓC(s) L(s)=((0.994−0.102i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3648s/2 ΓC(s) L(s)=((0.994−0.102i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36483648    =    26⋅3⋅192^{6} \cdot 3 \cdot 19
Sign: 0.994−0.102i0.994 - 0.102i
Analytic conductor: 1.820581.82058
Root analytic conductor: 1.349291.34929
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3648(2657,⋅)\chi_{3648} (2657, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3648, ( :0), 0.994−0.102i)(2,\ 3648,\ (\ :0),\ 0.994 - 0.102i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1085994821.108599482
L(12)L(\frac12) ≈\approx 1.1085994821.108599482
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+(0.342−0.939i)T 1 + (0.342 - 0.939i)T
19 1+(0.984−0.173i)T 1 + (0.984 - 0.173i)T
good5 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
7 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
11 1+(−0.342+0.592i)T+(−0.5−0.866i)T2 1 + (-0.342 + 0.592i)T + (-0.5 - 0.866i)T^{2}
13 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
17 1+(−1.11+1.32i)T+(−0.173−0.984i)T2 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2}
23 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
29 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(0.673+1.85i)T+(−0.766+0.642i)T2 1 + (0.673 + 1.85i)T + (-0.766 + 0.642i)T^{2}
43 1+(−0.984−0.173i)T+(0.939+0.342i)T2 1 + (-0.984 - 0.173i)T + (0.939 + 0.342i)T^{2}
47 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
53 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
59 1+(−0.524−0.439i)T+(0.173+0.984i)T2 1 + (-0.524 - 0.439i)T + (0.173 + 0.984i)T^{2}
61 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
67 1+(0.223+0.266i)T+(−0.173+0.984i)T2 1 + (0.223 + 0.266i)T + (-0.173 + 0.984i)T^{2}
71 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
73 1+(−1.76+0.642i)T+(0.766−0.642i)T2 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2}
79 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
83 1+(−0.984−1.70i)T+(−0.5+0.866i)T2 1 + (-0.984 - 1.70i)T + (-0.5 + 0.866i)T^{2}
89 1+(0.592−1.62i)T+(−0.766−0.642i)T2 1 + (0.592 - 1.62i)T + (-0.766 - 0.642i)T^{2}
97 1+(0.266+0.223i)T+(0.173+0.984i)T2 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)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

−8.884164969455163057215756407074, −8.164152900763519815826663399505, −7.15348015964899026017473006046, −6.44158803482475733172538469229, −5.50336553557779452634447457251, −5.09837537764155148317827452033, −4.04631334680109379550394919562, −3.41773408399465839820751644819, −2.47058596461490476184107435321, −0.804339778197589735333228525516, 1.14639102873515493502938980252, 2.04262007316300886589696236881, 3.05443777336045688085072335518, 4.15768360399769412074864631055, 5.00341499560778245170418347973, 5.94925101002878754183884946271, 6.45313140311597192840299175408, 7.17630719485387965949301279286, 7.992802947513609719414396457744, 8.454694471970046572291338816492

Graph of the ZZ-function along the critical line