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Properties

Label 2-3840-60.23-c0-0-0
Degree 22
Conductor 38403840
Sign 0.525−0.850i0.525 - 0.850i
Analytic cond. 1.916401.91640
Root an. cond. 1.384341.38434
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  + (−0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s + (−1 − i)7-s + 1.00i·9-s − 1.41·11-s + 1.00i·15-s + 1.41i·21-s + 1.00i·25-s + (0.707 − 0.707i)27-s − 1.41·29-s + (1.00 + 1.00i)33-s + 1.41i·35-s + (0.707 − 0.707i)45-s + i·49-s + (1.41 − 1.41i)53-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s + (−1 − i)7-s + 1.00i·9-s − 1.41·11-s + 1.00i·15-s + 1.41i·21-s + 1.00i·25-s + (0.707 − 0.707i)27-s − 1.41·29-s + (1.00 + 1.00i)33-s + 1.41i·35-s + (0.707 − 0.707i)45-s + i·49-s + (1.41 − 1.41i)53-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=((0.525−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s) L(s)=((0.525−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: 0.525−0.850i0.525 - 0.850i
Analytic conductor: 1.916401.91640
Root analytic conductor: 1.384341.38434
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3840(2303,⋅)\chi_{3840} (2303, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3840, ( :0), 0.525−0.850i)(2,\ 3840,\ (\ :0),\ 0.525 - 0.850i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.14189649970.1418964997
L(12)L(\frac12) ≈\approx 0.14189649970.1418964997
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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
5 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good7 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
11 1+1.41T+T2 1 + 1.41T + T^{2}
13 1+iT2 1 + iT^{2}
17 1+iT2 1 + iT^{2}
19 1+T2 1 + T^{2}
23 1+iT2 1 + iT^{2}
29 1+1.41T+T2 1 + 1.41T + T^{2}
31 1−T2 1 - T^{2}
37 1−iT2 1 - iT^{2}
41 1−T2 1 - T^{2}
43 1−iT2 1 - iT^{2}
47 1−iT2 1 - iT^{2}
53 1+(−1.41+1.41i)T−iT2 1 + (-1.41 + 1.41i)T - iT^{2}
59 1−1.41iT−T2 1 - 1.41iT - T^{2}
61 1+T2 1 + T^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
79 1+T2 1 + T^{2}
83 1+iT2 1 + iT^{2}
89 1+T2 1 + T^{2}
97 1+(1−i)T−iT2 1 + (1 - i)T - iT^{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

−8.567590256999021496499714759211, −7.84894924193560569017238493010, −7.32053554461701621740955088382, −6.76965564723351896655981742295, −5.71109969180679611295955930872, −5.18956083554128354429961786976, −4.24776924357716495338402229690, −3.44411546674206746126332692837, −2.28597611445949993884938697453, −0.915053585875729798530667157335, 0.11083515603850583116527908948, 2.38531818175852806652563158951, 3.14601656795937268743991551576, 3.83314349361459899363925217196, 4.86905661455494683552333515616, 5.63664737673803762005606925445, 6.16147523370628848105410247221, 7.01631656091592408685180313223, 7.73836138917316169352915302856, 8.643551936656958836002035600918

Graph of the ZZ-function along the critical line