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Properties

Label 2-1127-1.1-c1-0-63
Degree 22
Conductor 11271127
Sign −1-1
Analytic cond. 8.999148.99914
Root an. cond. 2.999852.99985
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.14·2-s − 2.43·3-s + 2.58·4-s − 0.818·5-s − 5.22·6-s + 1.24·8-s + 2.94·9-s − 1.75·10-s + 0.116·11-s − 6.30·12-s + 0.869·13-s + 1.99·15-s − 2.49·16-s + 0.498·17-s + 6.31·18-s − 4.92·19-s − 2.11·20-s + 0.249·22-s − 23-s − 3.04·24-s − 4.33·25-s + 1.86·26-s + 0.128·27-s − 7.25·29-s + 4.27·30-s − 8.02·31-s − 7.83·32-s + ⋯
L(s)  = 1  + 1.51·2-s − 1.40·3-s + 1.29·4-s − 0.366·5-s − 2.13·6-s + 0.441·8-s + 0.982·9-s − 0.554·10-s + 0.0350·11-s − 1.81·12-s + 0.241·13-s + 0.515·15-s − 0.623·16-s + 0.121·17-s + 1.48·18-s − 1.13·19-s − 0.472·20-s + 0.0531·22-s − 0.208·23-s − 0.621·24-s − 0.866·25-s + 0.365·26-s + 0.0246·27-s − 1.34·29-s + 0.780·30-s − 1.44·31-s − 1.38·32-s + ⋯

Functional equation

Λ(s)=(1127s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1127s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11271127    =    72⋅237^{2} \cdot 23
Sign: −1-1
Analytic conductor: 8.999148.99914
Root analytic conductor: 2.999852.99985
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1127, ( :1/2), −1)(2,\ 1127,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad7 1 1
23 1+T 1 + T
good2 1−2.14T+2T2 1 - 2.14T + 2T^{2}
3 1+2.43T+3T2 1 + 2.43T + 3T^{2}
5 1+0.818T+5T2 1 + 0.818T + 5T^{2}
11 1−0.116T+11T2 1 - 0.116T + 11T^{2}
13 1−0.869T+13T2 1 - 0.869T + 13T^{2}
17 1−0.498T+17T2 1 - 0.498T + 17T^{2}
19 1+4.92T+19T2 1 + 4.92T + 19T^{2}
29 1+7.25T+29T2 1 + 7.25T + 29T^{2}
31 1+8.02T+31T2 1 + 8.02T + 31T^{2}
37 1+7.51T+37T2 1 + 7.51T + 37T^{2}
41 1+6.61T+41T2 1 + 6.61T + 41T^{2}
43 1−10.5T+43T2 1 - 10.5T + 43T^{2}
47 1−1.36T+47T2 1 - 1.36T + 47T^{2}
53 1+7.53T+53T2 1 + 7.53T + 53T^{2}
59 1−10.0T+59T2 1 - 10.0T + 59T^{2}
61 1−6.31T+61T2 1 - 6.31T + 61T^{2}
67 1−5.05T+67T2 1 - 5.05T + 67T^{2}
71 1−9.70T+71T2 1 - 9.70T + 71T^{2}
73 1−8.78T+73T2 1 - 8.78T + 73T^{2}
79 1+0.541T+79T2 1 + 0.541T + 79T^{2}
83 1+3.81T+83T2 1 + 3.81T + 83T^{2}
89 1+17.9T+89T2 1 + 17.9T + 89T^{2}
97 1+16.2T+97T2 1 + 16.2T + 97T^{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

−9.641104038831044405818131199736, −8.490150394410519168707695652195, −7.23752296547079954049966966561, −6.55148268725148205705027740947, −5.68559972404170478623235329055, −5.29376329201043853464960404932, −4.21107902375552685857557457988, −3.61681780487672379605228489726, −2.03543401419923863021312601202, 0, 2.03543401419923863021312601202, 3.61681780487672379605228489726, 4.21107902375552685857557457988, 5.29376329201043853464960404932, 5.68559972404170478623235329055, 6.55148268725148205705027740947, 7.23752296547079954049966966561, 8.490150394410519168707695652195, 9.641104038831044405818131199736

Graph of the ZZ-function along the critical line