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Properties

Label 2-1100-1.1-c5-0-41
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  − 10.1·3-s + 237.·7-s − 139.·9-s + 121·11-s + 1.04e3·13-s + 1.98e3·17-s − 269.·19-s − 2.41e3·21-s + 1.94e3·23-s + 3.89e3·27-s − 801.·29-s + 7.82e3·31-s − 1.23e3·33-s + 1.30e4·37-s − 1.05e4·39-s − 4.29e3·41-s − 280.·43-s + 1.26e4·47-s + 3.97e4·49-s − 2.02e4·51-s − 3.44e4·53-s + 2.74e3·57-s − 5.10e3·59-s + 2.08e4·61-s − 3.31e4·63-s + 6.30e4·67-s − 1.97e4·69-s + ⋯
L(s)  = 1  − 0.652·3-s + 1.83·7-s − 0.574·9-s + 0.301·11-s + 1.70·13-s + 1.66·17-s − 0.171·19-s − 1.19·21-s + 0.764·23-s + 1.02·27-s − 0.176·29-s + 1.46·31-s − 0.196·33-s + 1.56·37-s − 1.11·39-s − 0.399·41-s − 0.0231·43-s + 0.833·47-s + 2.36·49-s − 1.08·51-s − 1.68·53-s + 0.111·57-s − 0.190·59-s + 0.718·61-s − 1.05·63-s + 1.71·67-s − 0.499·69-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 3.0886025483.088602548
L(12)L(\frac12) ≈\approx 3.0886025483.088602548
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+10.1T+243T2 1 + 10.1T + 243T^{2}
7 1−237.T+1.68e4T2 1 - 237.T + 1.68e4T^{2}
13 1−1.04e3T+3.71e5T2 1 - 1.04e3T + 3.71e5T^{2}
17 1−1.98e3T+1.41e6T2 1 - 1.98e3T + 1.41e6T^{2}
19 1+269.T+2.47e6T2 1 + 269.T + 2.47e6T^{2}
23 1−1.94e3T+6.43e6T2 1 - 1.94e3T + 6.43e6T^{2}
29 1+801.T+2.05e7T2 1 + 801.T + 2.05e7T^{2}
31 1−7.82e3T+2.86e7T2 1 - 7.82e3T + 2.86e7T^{2}
37 1−1.30e4T+6.93e7T2 1 - 1.30e4T + 6.93e7T^{2}
41 1+4.29e3T+1.15e8T2 1 + 4.29e3T + 1.15e8T^{2}
43 1+280.T+1.47e8T2 1 + 280.T + 1.47e8T^{2}
47 1−1.26e4T+2.29e8T2 1 - 1.26e4T + 2.29e8T^{2}
53 1+3.44e4T+4.18e8T2 1 + 3.44e4T + 4.18e8T^{2}
59 1+5.10e3T+7.14e8T2 1 + 5.10e3T + 7.14e8T^{2}
61 1−2.08e4T+8.44e8T2 1 - 2.08e4T + 8.44e8T^{2}
67 1−6.30e4T+1.35e9T2 1 - 6.30e4T + 1.35e9T^{2}
71 1+6.14e4T+1.80e9T2 1 + 6.14e4T + 1.80e9T^{2}
73 1+3.22e4T+2.07e9T2 1 + 3.22e4T + 2.07e9T^{2}
79 1+6.74e4T+3.07e9T2 1 + 6.74e4T + 3.07e9T^{2}
83 1+2.95e4T+3.93e9T2 1 + 2.95e4T + 3.93e9T^{2}
89 1−5.79e4T+5.58e9T2 1 - 5.79e4T + 5.58e9T^{2}
97 1+6.21e4T+8.58e9T2 1 + 6.21e4T + 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

−8.899204616808774298961103618640, −8.266773231894313904437798015816, −7.68527008062100310926578598483, −6.38977917778357720160932466668, −5.69149532861365780803269745657, −4.97159181804616336211727265655, −4.04069751326885656937289621458, −2.85699825054069552996458240751, −1.36998915119098787826402717179, −0.936333855439976301278171974006, 0.936333855439976301278171974006, 1.36998915119098787826402717179, 2.85699825054069552996458240751, 4.04069751326885656937289621458, 4.97159181804616336211727265655, 5.69149532861365780803269745657, 6.38977917778357720160932466668, 7.68527008062100310926578598483, 8.266773231894313904437798015816, 8.899204616808774298961103618640

Graph of the ZZ-function along the critical line