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Properties

Label 2-1386-77.76-c1-0-17
Degree 22
Conductor 13861386
Sign 0.983+0.181i0.983 + 0.181i
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 + 3.09i·5-s + (2.44 − i)7-s + i·8-s + 3.09·10-s + (1.79 − 2.79i)11-s + 0.646·13-s + (−1 − 2.44i)14-s + 16-s + 3.74·17-s − 1.80·19-s − 3.09i·20-s + (−2.79 − 1.79i)22-s − 4·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.38i·5-s + (0.925 − 0.377i)7-s + 0.353i·8-s + 0.978·10-s + (0.540 − 0.841i)11-s + 0.179·13-s + (−0.267 − 0.654i)14-s + 0.250·16-s + 0.907·17-s − 0.413·19-s − 0.692i·20-s + (−0.595 − 0.381i)22-s − 0.834·23-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 13861386    =    2⋅32⋅7⋅112 \cdot 3^{2} \cdot 7 \cdot 11
Sign: 0.983+0.181i0.983 + 0.181i
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), 0.983+0.181i)(2,\ 1386,\ (\ :1/2),\ 0.983 + 0.181i)

Particular Values

L(1)L(1) ≈\approx 1.8414463771.841446377
L(12)L(\frac12) ≈\approx 1.8414463771.841446377
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+(−1.79+2.79i)T 1 + (-1.79 + 2.79i)T
good5 1−3.09iT−5T2 1 - 3.09iT - 5T^{2}
13 1−0.646T+13T2 1 - 0.646T + 13T^{2}
17 1−3.74T+17T2 1 - 3.74T + 17T^{2}
19 1+1.80T+19T2 1 + 1.80T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−1.58iT−29T2 1 - 1.58iT - 29T^{2}
31 1−8.64iT−31T2 1 - 8.64iT - 31T^{2}
37 1−3.58T+37T2 1 - 3.58T + 37T^{2}
41 1−9.93T+41T2 1 - 9.93T + 41T^{2}
43 1−7.16iT−43T2 1 - 7.16iT - 43T^{2}
47 1+9.93iT−47T2 1 + 9.93iT - 47T^{2}
53 1−11.5T+53T2 1 - 11.5T + 53T^{2}
59 1+0.646iT−59T2 1 + 0.646iT - 59T^{2}
61 1−1.93T+61T2 1 - 1.93T + 61T^{2}
67 1+7.58T+67T2 1 + 7.58T + 67T^{2}
71 1+2T+71T2 1 + 2T + 71T^{2}
73 1−16.1T+73T2 1 - 16.1T + 73T^{2}
79 1−4iT−79T2 1 - 4iT - 79T^{2}
83 1+12.8T+83T2 1 + 12.8T + 83T^{2}
89 1−9.79iT−89T2 1 - 9.79iT - 89T^{2}
97 1−8.50iT−97T2 1 - 8.50iT - 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

−9.830026426971797215981268325377, −8.736266638810156945229863180902, −8.017902718413666291110799889173, −7.16347543025785488377431406072, −6.26843143484428845652848425835, −5.35414760262609230872059157172, −4.11202932288474047138367090809, −3.40794353248715910125853897215, −2.42626054674739687636906035675, −1.16096228776303297672964214644, 0.969545393288141126297892176315, 2.13687339910709612899241854598, 4.10652268999401654993347048353, 4.51273341886007975622237289294, 5.52095434584704191252006447440, 6.05849365707748282156146407547, 7.44409209947828515942336943122, 7.952108066750676008191235048685, 8.731440761961205795546182828538, 9.343420806576129491358655304142

Graph of the ZZ-function along the critical line