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Properties

Label 2-1872-12.11-c1-0-21
Degree 22
Conductor 18721872
Sign −0.577+0.816i-0.577 + 0.816i
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
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  − 2.82i·5-s − 4.47i·7-s + 3.16·11-s + 13-s − 4.24i·17-s + 6.32·23-s − 3.00·25-s − 4.24i·29-s + 8.94i·31-s − 12.6·35-s + 2·37-s − 5.65i·41-s + 3.16·47-s − 13.0·49-s − 1.41i·53-s + ⋯
L(s)  = 1  − 1.26i·5-s − 1.69i·7-s + 0.953·11-s + 0.277·13-s − 1.02i·17-s + 1.31·23-s − 0.600·25-s − 0.787i·29-s + 1.60i·31-s − 2.13·35-s + 0.328·37-s − 0.883i·41-s + 0.461·47-s − 1.85·49-s − 0.194i·53-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=((−0.577+0.816i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=((−0.577+0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: −0.577+0.816i-0.577 + 0.816i
Analytic conductor: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1872(287,⋅)\chi_{1872} (287, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1872, ( :1/2), −0.577+0.816i)(2,\ 1872,\ (\ :1/2),\ -0.577 + 0.816i)

Particular Values

L(1)L(1) ≈\approx 1.7965149121.796514912
L(12)L(\frac12) ≈\approx 1.7965149121.796514912
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 1
13 1−T 1 - T
good5 1+2.82iT−5T2 1 + 2.82iT - 5T^{2}
7 1+4.47iT−7T2 1 + 4.47iT - 7T^{2}
11 1−3.16T+11T2 1 - 3.16T + 11T^{2}
17 1+4.24iT−17T2 1 + 4.24iT - 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1−6.32T+23T2 1 - 6.32T + 23T^{2}
29 1+4.24iT−29T2 1 + 4.24iT - 29T^{2}
31 1−8.94iT−31T2 1 - 8.94iT - 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+5.65iT−41T2 1 + 5.65iT - 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1−3.16T+47T2 1 - 3.16T + 47T^{2}
53 1+1.41iT−53T2 1 + 1.41iT - 53T^{2}
59 1+9.48T+59T2 1 + 9.48T + 59T^{2}
61 1+61T2 1 + 61T^{2}
67 1−13.4iT−67T2 1 - 13.4iT - 67T^{2}
71 1−3.16T+71T2 1 - 3.16T + 71T^{2}
73 1+14T+73T2 1 + 14T + 73T^{2}
79 1−8.94iT−79T2 1 - 8.94iT - 79T^{2}
83 1+9.48T+83T2 1 + 9.48T + 83T^{2}
89 1+2.82iT−89T2 1 + 2.82iT - 89T^{2}
97 1+18T+97T2 1 + 18T + 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

−8.981513731593431123946396025003, −8.289465865685278602872873360175, −7.20312557556676882584703321066, −6.88053088984741477541700221424, −5.60827635027446955822564309875, −4.65837248489534732257726539707, −4.19178527797832428892309684761, −3.15870961881517777360578418214, −1.37208718953770828152417518222, −0.73893290627576563547112468838, 1.68842686424134164104966812611, 2.73065087346783331735260639501, 3.40666857042304315161270119194, 4.59035518778482000454234126683, 5.82478826561256160625582733371, 6.20776938673303346206010929869, 7.00277152457657592398163428887, 7.974548749807303477143611395277, 8.879896261903372271039386148429, 9.320851455203705509869657403938

Graph of the ZZ-function along the critical line