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Properties

Label 2-336-84.47-c0-0-0
Degree 22
Conductor 336336
Sign 0.605−0.795i0.605 - 0.795i
Analytic cond. 0.1676850.167685
Root an. cond. 0.4094940.409494
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 + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s − 1.73i·13-s + (0.5 − 0.866i)19-s − 0.999·21-s + (0.5 + 0.866i)25-s − 0.999·27-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (1.49 − 0.866i)39-s − 1.73i·43-s + (−0.499 − 0.866i)49-s + 0.999·57-s + (−0.499 − 0.866i)63-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s − 1.73i·13-s + (0.5 − 0.866i)19-s − 0.999·21-s + (0.5 + 0.866i)25-s − 0.999·27-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (1.49 − 0.866i)39-s − 1.73i·43-s + (−0.499 − 0.866i)49-s + 0.999·57-s + (−0.499 − 0.866i)63-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: 0.605−0.795i0.605 - 0.795i
Analytic conductor: 0.1676850.167685
Root analytic conductor: 0.4094940.409494
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ336(47,⋅)\chi_{336} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 336, ( :0), 0.605−0.795i)(2,\ 336,\ (\ :0),\ 0.605 - 0.795i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.87305904230.8730590423
L(12)L(\frac12) ≈\approx 0.87305904230.8730590423
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
7 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
11 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
13 1+1.73iT−T2 1 + 1.73iT - T^{2}
17 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
19 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
37 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1+1.73iT−T2 1 + 1.73iT - 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)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
71 1+T2 1 + T^{2}
73 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
79 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 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−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

−11.83470131765548927722160539289, −10.81322302568007167509701406376, −10.01000503464762470263496778463, −9.143574091934530282597870978785, −8.402858085987475224802568629153, −7.30692268328685899386654459694, −5.74976060019054884004783797835, −5.06907325214552342847721444665, −3.50072774304019219696254866375, −2.64751132197174822660784344246, 1.64775074292703857978510329863, 3.24457330532503732332431365700, 4.39197089541265841994041399687, 6.14864048087207485936368315123, 6.91443082553018309362574742675, 7.70496573747162889374351318459, 8.853429178655813035505718096920, 9.625824014948470037125116238875, 10.75648860056283010391157240057, 11.85532507274903725776029889146

Graph of the ZZ-function along the critical line