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Properties

Label 2-1872-1.1-c3-0-51
Degree 22
Conductor 18721872
Sign 11
Analytic cond. 110.451110.451
Root an. cond. 10.509510.5095
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20.2·5-s + 14.9·7-s + 55.0·11-s − 13·13-s − 83.5·17-s − 64.3·19-s − 21.1·23-s + 286.·25-s + 269.·29-s − 159.·31-s + 303.·35-s + 156.·37-s + 472.·41-s − 364.·43-s + 8.13·47-s − 119.·49-s + 640.·53-s + 1.11e3·55-s + 442.·59-s + 271.·61-s − 263.·65-s + 714.·67-s − 1.12e3·71-s + 425.·73-s + 822.·77-s − 12.3·79-s − 475.·83-s + ⋯
L(s)  = 1  + 1.81·5-s + 0.806·7-s + 1.50·11-s − 0.277·13-s − 1.19·17-s − 0.776·19-s − 0.191·23-s + 2.29·25-s + 1.72·29-s − 0.922·31-s + 1.46·35-s + 0.695·37-s + 1.79·41-s − 1.29·43-s + 0.0252·47-s − 0.349·49-s + 1.65·53-s + 2.73·55-s + 0.977·59-s + 0.570·61-s − 0.503·65-s + 1.30·67-s − 1.88·71-s + 0.682·73-s + 1.21·77-s − 0.0175·79-s − 0.628·83-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 110.451110.451
Root analytic conductor: 10.509510.5095
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1872, ( :3/2), 1)(2,\ 1872,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 4.1013838724.101383872
L(12)L(\frac12) ≈\approx 4.1013838724.101383872
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
3 1 1
13 1+13T 1 + 13T
good5 1−20.2T+125T2 1 - 20.2T + 125T^{2}
7 1−14.9T+343T2 1 - 14.9T + 343T^{2}
11 1−55.0T+1.33e3T2 1 - 55.0T + 1.33e3T^{2}
17 1+83.5T+4.91e3T2 1 + 83.5T + 4.91e3T^{2}
19 1+64.3T+6.85e3T2 1 + 64.3T + 6.85e3T^{2}
23 1+21.1T+1.21e4T2 1 + 21.1T + 1.21e4T^{2}
29 1−269.T+2.43e4T2 1 - 269.T + 2.43e4T^{2}
31 1+159.T+2.97e4T2 1 + 159.T + 2.97e4T^{2}
37 1−156.T+5.06e4T2 1 - 156.T + 5.06e4T^{2}
41 1−472.T+6.89e4T2 1 - 472.T + 6.89e4T^{2}
43 1+364.T+7.95e4T2 1 + 364.T + 7.95e4T^{2}
47 1−8.13T+1.03e5T2 1 - 8.13T + 1.03e5T^{2}
53 1−640.T+1.48e5T2 1 - 640.T + 1.48e5T^{2}
59 1−442.T+2.05e5T2 1 - 442.T + 2.05e5T^{2}
61 1−271.T+2.26e5T2 1 - 271.T + 2.26e5T^{2}
67 1−714.T+3.00e5T2 1 - 714.T + 3.00e5T^{2}
71 1+1.12e3T+3.57e5T2 1 + 1.12e3T + 3.57e5T^{2}
73 1−425.T+3.89e5T2 1 - 425.T + 3.89e5T^{2}
79 1+12.3T+4.93e5T2 1 + 12.3T + 4.93e5T^{2}
83 1+475.T+5.71e5T2 1 + 475.T + 5.71e5T^{2}
89 1−302.T+7.04e5T2 1 - 302.T + 7.04e5T^{2}
97 1+1.24e3T+9.12e5T2 1 + 1.24e3T + 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

−8.915777952794354009612025873827, −8.395696442935371946234438764199, −6.98672060998667312800834643042, −6.47556036209014060901459343212, −5.76663480079656310993450931466, −4.82575651737153552524485163280, −4.09268290159145792317328070068, −2.54591724813866756045013647559, −1.89696638219912390203250864842, −1.01471752348186714701707339569, 1.01471752348186714701707339569, 1.89696638219912390203250864842, 2.54591724813866756045013647559, 4.09268290159145792317328070068, 4.82575651737153552524485163280, 5.76663480079656310993450931466, 6.47556036209014060901459343212, 6.98672060998667312800834643042, 8.395696442935371946234438764199, 8.915777952794354009612025873827

Graph of the ZZ-function along the critical line