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Properties

Label 2-3840-40.29-c1-0-74
Degree 22
Conductor 38403840
Sign −0.948−0.316i-0.948 - 0.316i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (−2 + i)5-s − 4i·7-s + 9-s + 4i·11-s + (2 − i)15-s − 4i·17-s + 4i·21-s + 4i·23-s + (3 − 4i)25-s − 27-s − 6i·29-s − 4·31-s − 4i·33-s + (4 + 8i)35-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.894 + 0.447i)5-s − 1.51i·7-s + 0.333·9-s + 1.20i·11-s + (0.516 − 0.258i)15-s − 0.970i·17-s + 0.872i·21-s + 0.834i·23-s + (0.600 − 0.800i)25-s − 0.192·27-s − 1.11i·29-s − 0.718·31-s − 0.696i·33-s + (0.676 + 1.35i)35-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=((−0.948−0.316i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s+1/2) L(s)=((−0.948−0.316i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: −0.948−0.316i-0.948 - 0.316i
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3840(2689,⋅)\chi_{3840} (2689, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 3840, ( :1/2), −0.948−0.316i)(2,\ 3840,\ (\ :1/2),\ -0.948 - 0.316i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+T 1 + T
5 1+(2−i)T 1 + (2 - i)T
good7 1+4iT−7T2 1 + 4iT - 7T^{2}
11 1−4iT−11T2 1 - 4iT - 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+4iT−17T2 1 + 4iT - 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1+6iT−29T2 1 + 6iT - 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+8T+37T2 1 + 8T + 37T^{2}
41 1−10T+41T2 1 - 10T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+4iT−47T2 1 + 4iT - 47T^{2}
53 1−12T+53T2 1 - 12T + 53T^{2}
59 1−4iT−59T2 1 - 4iT - 59T^{2}
61 1+2iT−61T2 1 + 2iT - 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−8iT−73T2 1 - 8iT - 73T^{2}
79 1+12T+79T2 1 + 12T + 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+8iT−97T2 1 + 8iT - 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

−7.67100691982592494319962475267, −7.18213666900942543000580526143, −7.07665093602553087222345303659, −5.87723075347834717108006791312, −4.88361477114835320659974691443, −4.18130388667794124337835390943, −3.70967122958870468544864992828, −2.46040909099634949532930674713, −1.09743047650742575618685524963, 0, 1.32837498739201785889138248628, 2.60520460625860638316609689475, 3.53563637696182971539412512751, 4.36734315782402222249206983174, 5.39548168780312930976396048678, 5.72215104117866980872136179120, 6.55695265434507127195503621243, 7.47767137471417224354679673393, 8.407437464506144848470774426634

Graph of the ZZ-function along the critical line