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Properties

Label 2-1100-1.1-c5-0-21
Degree 22
Conductor 11001100
Sign 11
Analytic cond. 176.422176.422
Root an. cond. 13.282413.2824
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.8·3-s + 166.·7-s − 21.7·9-s + 121·11-s + 491.·13-s − 1.58e3·17-s + 172.·19-s − 2.47e3·21-s − 2.34e3·23-s + 3.93e3·27-s + 3.71e3·29-s + 7.94e3·31-s − 1.79e3·33-s + 3.46e3·37-s − 7.31e3·39-s + 1.25e4·41-s + 1.92e4·43-s − 2.76e4·47-s + 1.09e4·49-s + 2.35e4·51-s + 3.32e4·53-s − 2.56e3·57-s − 3.80e4·59-s − 6.56e3·61-s − 3.62e3·63-s − 6.00e4·67-s + 3.49e4·69-s + ⋯
L(s)  = 1  − 0.954·3-s + 1.28·7-s − 0.0895·9-s + 0.301·11-s + 0.807·13-s − 1.32·17-s + 0.109·19-s − 1.22·21-s − 0.926·23-s + 1.03·27-s + 0.819·29-s + 1.48·31-s − 0.287·33-s + 0.416·37-s − 0.770·39-s + 1.16·41-s + 1.58·43-s − 1.82·47-s + 0.649·49-s + 1.26·51-s + 1.62·53-s − 0.104·57-s − 1.42·59-s − 0.225·61-s − 0.115·63-s − 1.63·67-s + 0.883·69-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: 11
Analytic conductor: 176.422176.422
Root analytic conductor: 13.282413.2824
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1100, ( :5/2), 1)(2,\ 1100,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.7505210121.750521012
L(12)L(\frac12) ≈\approx 1.7505210121.750521012
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 1
5 1 1
11 1−121T 1 - 121T
good3 1+14.8T+243T2 1 + 14.8T + 243T^{2}
7 1−166.T+1.68e4T2 1 - 166.T + 1.68e4T^{2}
13 1−491.T+3.71e5T2 1 - 491.T + 3.71e5T^{2}
17 1+1.58e3T+1.41e6T2 1 + 1.58e3T + 1.41e6T^{2}
19 1−172.T+2.47e6T2 1 - 172.T + 2.47e6T^{2}
23 1+2.34e3T+6.43e6T2 1 + 2.34e3T + 6.43e6T^{2}
29 1−3.71e3T+2.05e7T2 1 - 3.71e3T + 2.05e7T^{2}
31 1−7.94e3T+2.86e7T2 1 - 7.94e3T + 2.86e7T^{2}
37 1−3.46e3T+6.93e7T2 1 - 3.46e3T + 6.93e7T^{2}
41 1−1.25e4T+1.15e8T2 1 - 1.25e4T + 1.15e8T^{2}
43 1−1.92e4T+1.47e8T2 1 - 1.92e4T + 1.47e8T^{2}
47 1+2.76e4T+2.29e8T2 1 + 2.76e4T + 2.29e8T^{2}
53 1−3.32e4T+4.18e8T2 1 - 3.32e4T + 4.18e8T^{2}
59 1+3.80e4T+7.14e8T2 1 + 3.80e4T + 7.14e8T^{2}
61 1+6.56e3T+8.44e8T2 1 + 6.56e3T + 8.44e8T^{2}
67 1+6.00e4T+1.35e9T2 1 + 6.00e4T + 1.35e9T^{2}
71 1+5.26e4T+1.80e9T2 1 + 5.26e4T + 1.80e9T^{2}
73 1+1.45e4T+2.07e9T2 1 + 1.45e4T + 2.07e9T^{2}
79 1+5.92e4T+3.07e9T2 1 + 5.92e4T + 3.07e9T^{2}
83 1−3.98e4T+3.93e9T2 1 - 3.98e4T + 3.93e9T^{2}
89 1+1.29e5T+5.58e9T2 1 + 1.29e5T + 5.58e9T^{2}
97 1−1.64e5T+8.58e9T2 1 - 1.64e5T + 8.58e9T^{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

−8.929704669057669938034998743683, −8.385321088091424797728130935036, −7.47864692820783619115644707375, −6.30461113518017143938901629316, −5.91978356561965489492213650809, −4.70412306344966319129672104691, −4.31211441432061659396914629450, −2.73854356549439115407926822824, −1.56549028518069819016539282236, −0.62673361611550052738025976653, 0.62673361611550052738025976653, 1.56549028518069819016539282236, 2.73854356549439115407926822824, 4.31211441432061659396914629450, 4.70412306344966319129672104691, 5.91978356561965489492213650809, 6.30461113518017143938901629316, 7.47864692820783619115644707375, 8.385321088091424797728130935036, 8.929704669057669938034998743683

Graph of the ZZ-function along the critical line