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Properties

Label 2-3840-8.5-c1-0-18
Degree 22
Conductor 38403840
Sign 0.707−0.707i0.707 - 0.707i
Analytic cond. 30.662530.6625
Root an. cond. 5.537375.53737
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − i·5-s − 2·7-s − 9-s − 2i·11-s + 2i·13-s + 15-s − 2·17-s − 2i·19-s − 2i·21-s − 2·23-s − 25-s − i·27-s + 6i·29-s + 4·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.447i·5-s − 0.755·7-s − 0.333·9-s − 0.603i·11-s + 0.554i·13-s + 0.258·15-s − 0.485·17-s − 0.458i·19-s − 0.436i·21-s − 0.417·23-s − 0.200·25-s − 0.192i·27-s + 1.11i·29-s + 0.718·31-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=((0.707−0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s+1/2) L(s)=((0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: 0.707−0.707i0.707 - 0.707i
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3840(1921,⋅)\chi_{3840} (1921, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3840, ( :1/2), 0.707−0.707i)(2,\ 3840,\ (\ :1/2),\ 0.707 - 0.707i)

Particular Values

L(1)L(1) ≈\approx 1.3831623841.383162384
L(12)L(\frac12) ≈\approx 1.3831623841.383162384
L(32)L(\frac{3}{2}) 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−iT 1 - iT
5 1+iT 1 + iT
good7 1+2T+7T2 1 + 2T + 7T^{2}
11 1+2iT−11T2 1 + 2iT - 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1+2iT−19T2 1 + 2iT - 19T^{2}
23 1+2T+23T2 1 + 2T + 23T^{2}
29 1−6iT−29T2 1 - 6iT - 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1−10T+41T2 1 - 10T + 41T^{2}
43 1+8iT−43T2 1 + 8iT - 43T^{2}
47 1−2T+47T2 1 - 2T + 47T^{2}
53 1−6iT−53T2 1 - 6iT - 53T^{2}
59 1−2iT−59T2 1 - 2iT - 59T^{2}
61 1−10iT−61T2 1 - 10iT - 61T^{2}
67 1+8iT−67T2 1 + 8iT - 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−16T+79T2 1 - 16T + 79T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+6T+97T2 1 + 6T + 97T^{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

−8.774952585729965456129835243451, −7.993912482476684355209898096027, −6.99942461829639485527240670252, −6.32539431110793658148666083920, −5.58083512635256841301766302033, −4.73637392265189121320659849815, −4.00343606549441224667644928499, −3.19988952236857052127572822872, −2.24930454546685715341295803470, −0.78859289201617975783698704764, 0.55929500020439937515470740784, 1.99369228606659514884963840121, 2.74643693783960879489484546063, 3.65577386419758656116920396697, 4.55047325233072525417247384495, 5.62232690846816836210592116059, 6.36100173357021849664235289123, 6.76742743840112987074920504934, 7.83256393604755001192482682697, 8.033509809642357855705635274075

Graph of the ZZ-function along the critical line