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Properties

Label 2-3648-228.179-c0-0-0
Degree 22
Conductor 36483648
Sign −0.813+0.582i-0.813 + 0.582i
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  + (−0.5 + 0.866i)3-s + 1.73i·7-s + (−0.499 − 0.866i)9-s + (−1.5 + 0.866i)13-s − 19-s + (−1.49 − 0.866i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s − 31-s − 1.73i·37-s − 1.73i·39-s + (1.5 + 0.866i)43-s − 1.99·49-s + (0.5 − 0.866i)57-s + (0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + 1.73i·7-s + (−0.499 − 0.866i)9-s + (−1.5 + 0.866i)13-s − 19-s + (−1.49 − 0.866i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s − 31-s − 1.73i·37-s − 1.73i·39-s + (1.5 + 0.866i)43-s − 1.99·49-s + (0.5 − 0.866i)57-s + (0.5 + 0.866i)61-s + ⋯

Functional equation

Λ(s)=(3648s/2 ΓC(s) L(s)=((−0.813+0.582i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3648s/2 ΓC(s) L(s)=((−0.813+0.582i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36483648    =    26⋅3⋅192^{6} \cdot 3 \cdot 19
Sign: −0.813+0.582i-0.813 + 0.582i
Analytic conductor: 1.820581.82058
Root analytic conductor: 1.349291.34929
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3648(2687,⋅)\chi_{3648} (2687, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3648, ( :0), −0.813+0.582i)(2,\ 3648,\ (\ :0),\ -0.813 + 0.582i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.35530398150.3553039815
L(12)L(\frac12) ≈\approx 0.35530398150.3553039815
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
19 1+T 1 + T
good5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
7 1−1.73iT−T2 1 - 1.73iT - T^{2}
11 1+T2 1 + T^{2}
13 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
29 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
31 1+T+T2 1 + T + T^{2}
37 1+1.73iT−T2 1 + 1.73iT - T^{2}
41 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
43 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}
47 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
53 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
97 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)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

−9.153289124038776760781314716877, −8.768516581872050499622148219865, −7.74295182740970361937226180150, −6.76450412791501293378726954671, −5.95010641354179295038178320084, −5.47493592198504784068653974027, −4.63266400249068039195423003787, −3.98676955490813340979976280926, −2.66495388798721724046987483415, −2.14300487645527394671849981937, 0.21034675185649874579726952458, 1.41093571231103312256063047175, 2.50607419701215875357456209617, 3.60205342973356493811357115760, 4.56998241593209598537165650183, 5.24019071252309200767201497345, 6.17758762655377163928460390401, 6.98118951329108797336250656207, 7.50564768964677266216720850221, 7.86175620137304296342017657210

Graph of the ZZ-function along the critical line