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Properties

Label 2-1386-77.76-c1-0-36
Degree 22
Conductor 13861386
Sign −0.818−0.575i-0.818 - 0.575i
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)=((−0.818−0.575i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1386s/2 ΓC(s+1/2) L(s)=((−0.818−0.575i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13861386    =    2⋅32⋅7⋅112 \cdot 3^{2} \cdot 7 \cdot 11
Sign: −0.818−0.575i-0.818 - 0.575i
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), −0.818−0.575i)(2,\ 1386,\ (\ :1/2),\ -0.818 - 0.575i)

Particular Values

L(1)L(1) ≈\approx 0.34848020640.3484802064
L(12)L(\frac12) ≈\approx 0.34848020640.3484802064
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

−9.078198593159204957625293071808, −8.388997575106378603937073990968, −7.65544020786271130556750879237, −6.73993079734012989180684356485, −5.43889134487594145638606215481, −4.65228969717493853598518622416, −4.10009880352403347777561178965, −2.53761138395734820327141860571, −1.83718892309898971538699177901, −0.13063235393916447917351413837, 1.92430273093018105522092504308, 3.03684669497185071045077912079, 4.46816497701266335204399505408, 5.03468510411286841875783342983, 5.96690444497082575979600804959, 6.84472729598756716024855387930, 7.62480072870278002279339497565, 8.500714154532331950852086135198, 8.844625311322821311704352027082, 10.10975072242941527611152781429

Graph of the ZZ-function along the critical line