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Properties

Label 2-1472-368.45-c0-0-0
Degree 22
Conductor 14721472
Sign 0.130−0.991i0.130 - 0.991i
Analytic cond. 0.7346230.734623
Root an. cond. 0.8571010.857101
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 1.36i)3-s + 2.73i·9-s + (−0.366 − 0.366i)13-s − i·23-s − i·25-s + (−2.36 + 2.36i)27-s + (1.36 + 1.36i)29-s − 1.73·31-s − i·39-s − i·41-s + 47-s − 49-s + (1 − i)59-s + (1.36 − 1.36i)69-s − 1.73i·71-s + ⋯
L(s)  = 1  + (1.36 + 1.36i)3-s + 2.73i·9-s + (−0.366 − 0.366i)13-s − i·23-s − i·25-s + (−2.36 + 2.36i)27-s + (1.36 + 1.36i)29-s − 1.73·31-s − i·39-s − i·41-s + 47-s − 49-s + (1 − i)59-s + (1.36 − 1.36i)69-s − 1.73i·71-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=((0.130−0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s) L(s)=((0.130−0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: 0.130−0.991i0.130 - 0.991i
Analytic conductor: 0.7346230.734623
Root analytic conductor: 0.8571010.857101
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1472(1425,⋅)\chi_{1472} (1425, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1472, ( :0), 0.130−0.991i)(2,\ 1472,\ (\ :0),\ 0.130 - 0.991i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6582570561.658257056
L(12)L(\frac12) ≈\approx 1.6582570561.658257056
L(1)L(1) 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
23 1+iT 1 + iT
good3 1+(−1.36−1.36i)T+iT2 1 + (-1.36 - 1.36i)T + iT^{2}
5 1+iT2 1 + iT^{2}
7 1+T2 1 + T^{2}
11 1+iT2 1 + iT^{2}
13 1+(0.366+0.366i)T+iT2 1 + (0.366 + 0.366i)T + iT^{2}
17 1−T2 1 - T^{2}
19 1−iT2 1 - iT^{2}
29 1+(−1.36−1.36i)T+iT2 1 + (-1.36 - 1.36i)T + iT^{2}
31 1+1.73T+T2 1 + 1.73T + T^{2}
37 1+iT2 1 + iT^{2}
41 1+iT−T2 1 + iT - T^{2}
43 1+iT2 1 + iT^{2}
47 1−T+T2 1 - T + T^{2}
53 1+iT2 1 + iT^{2}
59 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
61 1−iT2 1 - iT^{2}
67 1−iT2 1 - iT^{2}
71 1+1.73iT−T2 1 + 1.73iT - T^{2}
73 1−iT−T2 1 - iT - T^{2}
79 1−T2 1 - T^{2}
83 1−iT2 1 - iT^{2}
89 1+T2 1 + T^{2}
97 1−T2 1 - T^{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

−9.825381591052167058075017732320, −9.028987984145465172044491161454, −8.508089998849182011810588266819, −7.79084639925570971761793109776, −6.81040914004167357056579837348, −5.39929587373681317054063679733, −4.70171025933114825966281854797, −3.85154042028488456270290057594, −3.00622692088938132002009658678, −2.14950372428015083013317138813, 1.33310930487965156410094574986, 2.30341736868639320295973446477, 3.20350184328983946501665285744, 4.12724559313902438202457951295, 5.59911715850064338962734609561, 6.56373733740205735566123441484, 7.29049630465938043438290388362, 7.81652026419501100265551175732, 8.631457448103184264052196679000, 9.314748864651358753968177033386

Graph of the ZZ-function along the critical line