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Properties

Label 2-1638-13.12-c1-0-12
Degree 22
Conductor 16381638
Sign 0.907−0.419i0.907 - 0.419i
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
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·2-s − 4-s + 2.54i·5-s − i·7-s + i·8-s + 2.54·10-s − 3.33i·11-s + (−3.27 + 1.51i)13-s − 14-s + 16-s + 7.09·17-s + 4.54i·19-s − 2.54i·20-s − 3.33·22-s + 1.75·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.13i·5-s − 0.377i·7-s + 0.353i·8-s + 0.804·10-s − 1.00i·11-s + (−0.907 + 0.419i)13-s − 0.267·14-s + 0.250·16-s + 1.71·17-s + 1.04i·19-s − 0.569i·20-s − 0.710·22-s + 0.366·23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((0.907−0.419i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((0.907−0.419i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.907−0.419i0.907 - 0.419i
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1638(883,⋅)\chi_{1638} (883, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1638, ( :1/2), 0.907−0.419i)(2,\ 1638,\ (\ :1/2),\ 0.907 - 0.419i)

Particular Values

L(1)L(1) ≈\approx 1.4175341311.417534131
L(12)L(\frac12) ≈\approx 1.4175341311.417534131
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+iT 1 + iT
3 1 1
7 1+iT 1 + iT
13 1+(3.27−1.51i)T 1 + (3.27 - 1.51i)T
good5 1−2.54iT−5T2 1 - 2.54iT - 5T^{2}
11 1+3.33iT−11T2 1 + 3.33iT - 11T^{2}
17 1−7.09T+17T2 1 - 7.09T + 17T^{2}
19 1−4.54iT−19T2 1 - 4.54iT - 19T^{2}
23 1−1.75T+23T2 1 - 1.75T + 23T^{2}
29 1+7.57T+29T2 1 + 7.57T + 29T^{2}
31 1−3.33iT−31T2 1 - 3.33iT - 31T^{2}
37 1+3.75iT−37T2 1 + 3.75iT - 37T^{2}
41 1−4.24iT−41T2 1 - 4.24iT - 41T^{2}
43 1−9.09T+43T2 1 - 9.09T + 43T^{2}
47 1−11.3iT−47T2 1 - 11.3iT - 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1−8.06iT−59T2 1 - 8.06iT - 59T^{2}
61 1+0.785T+61T2 1 + 0.785T + 61T^{2}
67 1−5.75iT−67T2 1 - 5.75iT - 67T^{2}
71 1−11.1iT−71T2 1 - 11.1iT - 71T^{2}
73 1−9.33iT−73T2 1 - 9.33iT - 73T^{2}
79 1−4.85T+79T2 1 - 4.85T + 79T^{2}
83 1−11.2iT−83T2 1 - 11.2iT - 83T^{2}
89 1+15.0iT−89T2 1 + 15.0iT - 89T^{2}
97 1+7.75iT−97T2 1 + 7.75iT - 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

−9.729335220416505249324305098991, −8.781814089037556796131262013298, −7.68260955679953935826521609466, −7.28309422848531062502652433407, −6.06770248878088331666782537238, −5.41577668338039580885831538896, −4.08143528449735087195759981861, −3.33079501071606263418679405479, −2.58507684435035833590988941521, −1.18063557624227386495500910813, 0.63019109724969909408816700856, 2.10548935790796452728326761600, 3.51253317612974734145963209624, 4.74364796271819359476741477701, 5.14157187608412471362582694543, 5.90750953326133879694370515052, 7.18639270060927706581343487475, 7.61708007622658875900182328421, 8.497419175252056365387965448392, 9.364583494257564269985298765449

Graph of the ZZ-function along the critical line