↑

Properties

Label 2-882-1.1-c5-0-5
Degree 22
Conductor 882882
Sign 11
Analytic cond. 141.458141.458
Root an. cond. 11.893611.8936
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 16·4-s − 61.0·5-s − 64·8-s + 244.·10-s + 36.5·11-s − 34.5·13-s + 256·16-s − 2.06e3·17-s − 452.·19-s − 977.·20-s − 146.·22-s − 1.68e3·23-s + 604.·25-s + 138.·26-s + 4.76e3·29-s + 5.26e3·31-s − 1.02e3·32-s + 8.24e3·34-s − 1.28e4·37-s + 1.80e3·38-s + 3.90e3·40-s − 7.12e3·41-s + 1.11e4·43-s + 584.·44-s + 6.73e3·46-s − 2.34e4·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.09·5-s − 0.353·8-s + 0.772·10-s + 0.0910·11-s − 0.0567·13-s + 0.250·16-s − 1.72·17-s − 0.287·19-s − 0.546·20-s − 0.0643·22-s − 0.663·23-s + 0.193·25-s + 0.0401·26-s + 1.05·29-s + 0.983·31-s − 0.176·32-s + 1.22·34-s − 1.53·37-s + 0.203·38-s + 0.386·40-s − 0.662·41-s + 0.918·43-s + 0.0455·44-s + 0.469·46-s − 1.54·47-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 141.458141.458
Root analytic conductor: 11.893611.8936
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 882, ( :5/2), 1)(2,\ 882,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.44629974180.4462997418
L(12)L(\frac12) ≈\approx 0.44629974180.4462997418
L(72)L(\frac{7}{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+4T 1 + 4T
3 1 1
7 1 1
good5 1+61.0T+3.12e3T2 1 + 61.0T + 3.12e3T^{2}
11 1−36.5T+1.61e5T2 1 - 36.5T + 1.61e5T^{2}
13 1+34.5T+3.71e5T2 1 + 34.5T + 3.71e5T^{2}
17 1+2.06e3T+1.41e6T2 1 + 2.06e3T + 1.41e6T^{2}
19 1+452.T+2.47e6T2 1 + 452.T + 2.47e6T^{2}
23 1+1.68e3T+6.43e6T2 1 + 1.68e3T + 6.43e6T^{2}
29 1−4.76e3T+2.05e7T2 1 - 4.76e3T + 2.05e7T^{2}
31 1−5.26e3T+2.86e7T2 1 - 5.26e3T + 2.86e7T^{2}
37 1+1.28e4T+6.93e7T2 1 + 1.28e4T + 6.93e7T^{2}
41 1+7.12e3T+1.15e8T2 1 + 7.12e3T + 1.15e8T^{2}
43 1−1.11e4T+1.47e8T2 1 - 1.11e4T + 1.47e8T^{2}
47 1+2.34e4T+2.29e8T2 1 + 2.34e4T + 2.29e8T^{2}
53 1−7.03e3T+4.18e8T2 1 - 7.03e3T + 4.18e8T^{2}
59 1+4.42e4T+7.14e8T2 1 + 4.42e4T + 7.14e8T^{2}
61 1+1.93e4T+8.44e8T2 1 + 1.93e4T + 8.44e8T^{2}
67 1−2.09e4T+1.35e9T2 1 - 2.09e4T + 1.35e9T^{2}
71 1+7.98e4T+1.80e9T2 1 + 7.98e4T + 1.80e9T^{2}
73 1−3.70e4T+2.07e9T2 1 - 3.70e4T + 2.07e9T^{2}
79 1−4.20e4T+3.07e9T2 1 - 4.20e4T + 3.07e9T^{2}
83 1+6.31e3T+3.93e9T2 1 + 6.31e3T + 3.93e9T^{2}
89 1+5.14e4T+5.58e9T2 1 + 5.14e4T + 5.58e9T^{2}
97 1−1.27e5T+8.58e9T2 1 - 1.27e5T + 8.58e9T^{2}
show more
show less
   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

−9.235278925991646510597263529544, −8.475161971037858044456120283305, −7.907157226486664259676548712457, −6.91369447500939324128543223282, −6.29119698402347659698462831588, −4.81872773617877933496246031093, −4.00686108909138917378106158666, −2.86829057405911945196047649551, −1.71387450032464521830873079177, −0.32547991289970319359220750820, 0.32547991289970319359220750820, 1.71387450032464521830873079177, 2.86829057405911945196047649551, 4.00686108909138917378106158666, 4.81872773617877933496246031093, 6.29119698402347659698462831588, 6.91369447500939324128543223282, 7.907157226486664259676548712457, 8.475161971037858044456120283305, 9.235278925991646510597263529544

Graph of the ZZ-function along the critical line