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Properties

Label 2-1386-77.76-c1-0-10
Degree 22
Conductor 13861386
Sign 0.181−0.983i0.181 - 0.983i
Analytic cond. 11.067211.0672
Root an. cond. 3.326753.32675
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 − 0.646i·5-s + (−2.44 + i)7-s − i·8-s + 0.646·10-s + (−2.79 − 1.79i)11-s + 3.09·13-s + (−1 − 2.44i)14-s + 16-s + 3.74·17-s + 5.54·19-s + 0.646i·20-s + (1.79 − 2.79i)22-s − 4·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.288i·5-s + (−0.925 + 0.377i)7-s − 0.353i·8-s + 0.204·10-s + (−0.841 − 0.540i)11-s + 0.858·13-s + (−0.267 − 0.654i)14-s + 0.250·16-s + 0.907·17-s + 1.27·19-s + 0.144i·20-s + (0.381 − 0.595i)22-s − 0.834·23-s + ⋯

Functional equation

Λ(s)=(1386s/2 ΓC(s) L(s)=((0.181−0.983i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1386s/2 ΓC(s+1/2) L(s)=((0.181−0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13861386    =    2⋅32⋅7⋅112 \cdot 3^{2} \cdot 7 \cdot 11
Sign: 0.181−0.983i0.181 - 0.983i
Analytic conductor: 11.067211.0672
Root analytic conductor: 3.326753.32675
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1386(307,⋅)\chi_{1386} (307, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1386, ( :1/2), 0.181−0.983i)(2,\ 1386,\ (\ :1/2),\ 0.181 - 0.983i)

Particular Values

L(1)L(1) ≈\approx 1.3217558261.321755826
L(12)L(\frac12) ≈\approx 1.3217558261.321755826
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+(2.44−i)T 1 + (2.44 - i)T
11 1+(2.79+1.79i)T 1 + (2.79 + 1.79i)T
good5 1+0.646iT−5T2 1 + 0.646iT - 5T^{2}
13 1−3.09T+13T2 1 - 3.09T + 13T^{2}
17 1−3.74T+17T2 1 - 3.74T + 17T^{2}
19 1−5.54T+19T2 1 - 5.54T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−7.58iT−29T2 1 - 7.58iT - 29T^{2}
31 1−1.15iT−31T2 1 - 1.15iT - 31T^{2}
37 1+5.58T+37T2 1 + 5.58T + 37T^{2}
41 1−5.03T+41T2 1 - 5.03T + 41T^{2}
43 1−11.1iT−43T2 1 - 11.1iT - 43T^{2}
47 1−5.03iT−47T2 1 - 5.03iT - 47T^{2}
53 1−2.41T+53T2 1 - 2.41T + 53T^{2}
59 1−3.09iT−59T2 1 - 3.09iT - 59T^{2}
61 1−9.28T+61T2 1 - 9.28T + 61T^{2}
67 1−1.58T+67T2 1 - 1.58T + 67T^{2}
71 1+2T+71T2 1 + 2T + 71T^{2}
73 1−6.32T+73T2 1 - 6.32T + 73T^{2}
79 1+4iT−79T2 1 + 4iT - 79T^{2}
83 1−9.15T+83T2 1 - 9.15T + 83T^{2}
89 1−9.79iT−89T2 1 - 9.79iT - 89T^{2}
97 1−15.9iT−97T2 1 - 15.9iT - 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

−9.593362763762765886235443742670, −8.854476416169683288136895372634, −8.144349036230312957186600809772, −7.33778460813291288732230336968, −6.38991051307602661919373967885, −5.64941013223157986264846239999, −5.03578724969221027350554088949, −3.62711239940977923416926413841, −2.94557002962312817736206494156, −1.03241790800917581745856716717, 0.69457561259195189409199766367, 2.23684250662594339649195938343, 3.28403455345688970922033534605, 3.92260474534398067938297659122, 5.19422021301334545663294515132, 5.97826120838558245981870547394, 7.06116633905250795341151243492, 7.78238794599527531614589292270, 8.726015165121194172135476407321, 9.756898723467485391864483258333

Graph of the ZZ-function along the critical line