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Properties

Label 2-1472-1.1-c3-0-109
Degree 22
Conductor 14721472
Sign −1-1
Analytic cond. 86.850886.8508
Root an. cond. 9.319379.31937
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.24·3-s + 5.01·5-s − 22.4·7-s − 16.4·9-s + 50.5·11-s + 89.2·13-s + 16.2·15-s − 66.9·17-s − 97.7·19-s − 72.9·21-s − 23·23-s − 99.8·25-s − 141.·27-s − 29.9·29-s − 73.5·31-s + 163.·33-s − 112.·35-s − 1.12·37-s + 289.·39-s + 34.3·41-s + 160.·43-s − 82.7·45-s − 289.·47-s + 162.·49-s − 217.·51-s + 392.·53-s + 253.·55-s + ⋯
L(s)  = 1  + 0.624·3-s + 0.448·5-s − 1.21·7-s − 0.610·9-s + 1.38·11-s + 1.90·13-s + 0.280·15-s − 0.954·17-s − 1.17·19-s − 0.757·21-s − 0.208·23-s − 0.798·25-s − 1.00·27-s − 0.192·29-s − 0.426·31-s + 0.864·33-s − 0.545·35-s − 0.00500·37-s + 1.18·39-s + 0.130·41-s + 0.569·43-s − 0.273·45-s − 0.898·47-s + 0.474·49-s − 0.595·51-s + 1.01·53-s + 0.621·55-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: −1-1
Analytic conductor: 86.850886.8508
Root analytic conductor: 9.319379.31937
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1472, ( :3/2), −1)(2,\ 1472,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
23 1+23T 1 + 23T
good3 1−3.24T+27T2 1 - 3.24T + 27T^{2}
5 1−5.01T+125T2 1 - 5.01T + 125T^{2}
7 1+22.4T+343T2 1 + 22.4T + 343T^{2}
11 1−50.5T+1.33e3T2 1 - 50.5T + 1.33e3T^{2}
13 1−89.2T+2.19e3T2 1 - 89.2T + 2.19e3T^{2}
17 1+66.9T+4.91e3T2 1 + 66.9T + 4.91e3T^{2}
19 1+97.7T+6.85e3T2 1 + 97.7T + 6.85e3T^{2}
29 1+29.9T+2.43e4T2 1 + 29.9T + 2.43e4T^{2}
31 1+73.5T+2.97e4T2 1 + 73.5T + 2.97e4T^{2}
37 1+1.12T+5.06e4T2 1 + 1.12T + 5.06e4T^{2}
41 1−34.3T+6.89e4T2 1 - 34.3T + 6.89e4T^{2}
43 1−160.T+7.95e4T2 1 - 160.T + 7.95e4T^{2}
47 1+289.T+1.03e5T2 1 + 289.T + 1.03e5T^{2}
53 1−392.T+1.48e5T2 1 - 392.T + 1.48e5T^{2}
59 1−174.T+2.05e5T2 1 - 174.T + 2.05e5T^{2}
61 1−474.T+2.26e5T2 1 - 474.T + 2.26e5T^{2}
67 1+671.T+3.00e5T2 1 + 671.T + 3.00e5T^{2}
71 1−481.T+3.57e5T2 1 - 481.T + 3.57e5T^{2}
73 1+778.T+3.89e5T2 1 + 778.T + 3.89e5T^{2}
79 1−132.T+4.93e5T2 1 - 132.T + 4.93e5T^{2}
83 1+808.T+5.71e5T2 1 + 808.T + 5.71e5T^{2}
89 1+1.05e3T+7.04e5T2 1 + 1.05e3T + 7.04e5T^{2}
97 1−824.T+9.12e5T2 1 - 824.T + 9.12e5T^{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

−8.887195605291232101115516767968, −8.249958545733039998493398311219, −6.87306210111826920794933608549, −6.25756508616341477133847366171, −5.80388754757634618470750470647, −4.03630209144009521237940898566, −3.67598911253509922370088794556, −2.54145687691683855120552076672, −1.49183817081434813619177605511, 0, 1.49183817081434813619177605511, 2.54145687691683855120552076672, 3.67598911253509922370088794556, 4.03630209144009521237940898566, 5.80388754757634618470750470647, 6.25756508616341477133847366171, 6.87306210111826920794933608549, 8.249958545733039998493398311219, 8.887195605291232101115516767968

Graph of the ZZ-function along the critical line