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Properties

Label 2-1100-1.1-c5-0-32
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  + 21.2·3-s − 140.·7-s + 206.·9-s + 121·11-s + 1.07e3·13-s − 2.02e3·17-s + 79.8·19-s − 2.98e3·21-s + 96.0·23-s − 764.·27-s + 1.69e3·29-s + 2.33e3·31-s + 2.56e3·33-s + 6.71e3·37-s + 2.27e4·39-s + 1.99e4·41-s + 3.48e3·43-s + 7.98e3·47-s + 3.06e3·49-s − 4.28e4·51-s − 3.28e4·53-s + 1.69e3·57-s − 4.79e4·59-s + 1.99e4·61-s − 2.91e4·63-s + 2.24e4·67-s + 2.03e3·69-s + ⋯
L(s)  = 1  + 1.36·3-s − 1.08·7-s + 0.851·9-s + 0.301·11-s + 1.75·13-s − 1.69·17-s + 0.0507·19-s − 1.47·21-s + 0.0378·23-s − 0.201·27-s + 0.373·29-s + 0.435·31-s + 0.410·33-s + 0.806·37-s + 2.39·39-s + 1.85·41-s + 0.287·43-s + 0.527·47-s + 0.182·49-s − 2.30·51-s − 1.60·53-s + 0.0690·57-s − 1.79·59-s + 0.685·61-s − 0.925·63-s + 0.611·67-s + 0.0515·69-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 3.5248035343.524803534
L(12)L(\frac12) ≈\approx 3.5248035343.524803534
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 1−21.2T+243T2 1 - 21.2T + 243T^{2}
7 1+140.T+1.68e4T2 1 + 140.T + 1.68e4T^{2}
13 1−1.07e3T+3.71e5T2 1 - 1.07e3T + 3.71e5T^{2}
17 1+2.02e3T+1.41e6T2 1 + 2.02e3T + 1.41e6T^{2}
19 1−79.8T+2.47e6T2 1 - 79.8T + 2.47e6T^{2}
23 1−96.0T+6.43e6T2 1 - 96.0T + 6.43e6T^{2}
29 1−1.69e3T+2.05e7T2 1 - 1.69e3T + 2.05e7T^{2}
31 1−2.33e3T+2.86e7T2 1 - 2.33e3T + 2.86e7T^{2}
37 1−6.71e3T+6.93e7T2 1 - 6.71e3T + 6.93e7T^{2}
41 1−1.99e4T+1.15e8T2 1 - 1.99e4T + 1.15e8T^{2}
43 1−3.48e3T+1.47e8T2 1 - 3.48e3T + 1.47e8T^{2}
47 1−7.98e3T+2.29e8T2 1 - 7.98e3T + 2.29e8T^{2}
53 1+3.28e4T+4.18e8T2 1 + 3.28e4T + 4.18e8T^{2}
59 1+4.79e4T+7.14e8T2 1 + 4.79e4T + 7.14e8T^{2}
61 1−1.99e4T+8.44e8T2 1 - 1.99e4T + 8.44e8T^{2}
67 1−2.24e4T+1.35e9T2 1 - 2.24e4T + 1.35e9T^{2}
71 1−9.67e3T+1.80e9T2 1 - 9.67e3T + 1.80e9T^{2}
73 1−2.07e4T+2.07e9T2 1 - 2.07e4T + 2.07e9T^{2}
79 1−9.56e4T+3.07e9T2 1 - 9.56e4T + 3.07e9T^{2}
83 1−2.45e4T+3.93e9T2 1 - 2.45e4T + 3.93e9T^{2}
89 1−6.79e4T+5.58e9T2 1 - 6.79e4T + 5.58e9T^{2}
97 1−7.60e4T+8.58e9T2 1 - 7.60e4T + 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

−9.164643908543061920431643253024, −8.459381356217162728549569921863, −7.67397776338948057449731722160, −6.50034356767374029387290476495, −6.12946618679861536854256726970, −4.44516207644179983788323522076, −3.67818527367736416934057964646, −2.93866479704281369527279350870, −2.02184468734267250435254703291, −0.74209466607610779834925408269, 0.74209466607610779834925408269, 2.02184468734267250435254703291, 2.93866479704281369527279350870, 3.67818527367736416934057964646, 4.44516207644179983788323522076, 6.12946618679861536854256726970, 6.50034356767374029387290476495, 7.67397776338948057449731722160, 8.459381356217162728549569921863, 9.164643908543061920431643253024

Graph of the ZZ-function along the critical line