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Properties

Label 2-1127-161.101-c0-0-0
Degree 22
Conductor 11271127
Sign −0.244−0.969i-0.244 - 0.969i
Analytic cond. 0.5624460.562446
Root an. cond. 0.7499640.749964
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.21 + 1.16i)2-s + (0.0871 + 1.82i)4-s + (−0.915 + 1.05i)8-s + (0.928 − 0.371i)9-s + (−0.947 + 0.903i)11-s + (−0.518 + 0.0495i)16-s + (1.56 + 0.625i)18-s − 2.20·22-s + (−0.786 + 0.618i)23-s + (0.235 − 0.971i)25-s + (−1.61 − 1.03i)29-s + (0.409 + 0.322i)32-s + (0.760 + 1.66i)36-s + (0.771 − 0.308i)37-s + (0.186 + 0.215i)43-s + (−1.73 − 1.65i)44-s + ⋯
L(s)  = 1  + (1.21 + 1.16i)2-s + (0.0871 + 1.82i)4-s + (−0.915 + 1.05i)8-s + (0.928 − 0.371i)9-s + (−0.947 + 0.903i)11-s + (−0.518 + 0.0495i)16-s + (1.56 + 0.625i)18-s − 2.20·22-s + (−0.786 + 0.618i)23-s + (0.235 − 0.971i)25-s + (−1.61 − 1.03i)29-s + (0.409 + 0.322i)32-s + (0.760 + 1.66i)36-s + (0.771 − 0.308i)37-s + (0.186 + 0.215i)43-s + (−1.73 − 1.65i)44-s + ⋯

Functional equation

Λ(s)=(1127s/2 ΓC(s) L(s)=((−0.244−0.969i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1127s/2 ΓC(s) L(s)=((−0.244−0.969i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11271127    =    72⋅237^{2} \cdot 23
Sign: −0.244−0.969i-0.244 - 0.969i
Analytic conductor: 0.5624460.562446
Root analytic conductor: 0.7499640.749964
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1127(423,⋅)\chi_{1127} (423, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1127, ( :0), −0.244−0.969i)(2,\ 1127,\ (\ :0),\ -0.244 - 0.969i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9387840321.938784032
L(12)L(\frac12) ≈\approx 1.9387840321.938784032
L(1)L(1) 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+(0.786−0.618i)T 1 + (0.786 - 0.618i)T
good2 1+(−1.21−1.16i)T+(0.0475+0.998i)T2 1 + (-1.21 - 1.16i)T + (0.0475 + 0.998i)T^{2}
3 1+(−0.928+0.371i)T2 1 + (-0.928 + 0.371i)T^{2}
5 1+(−0.235+0.971i)T2 1 + (-0.235 + 0.971i)T^{2}
11 1+(0.947−0.903i)T+(0.0475−0.998i)T2 1 + (0.947 - 0.903i)T + (0.0475 - 0.998i)T^{2}
13 1+(0.654−0.755i)T2 1 + (0.654 - 0.755i)T^{2}
17 1+(−0.580+0.814i)T2 1 + (-0.580 + 0.814i)T^{2}
19 1+(−0.580−0.814i)T2 1 + (-0.580 - 0.814i)T^{2}
29 1+(1.61+1.03i)T+(0.415+0.909i)T2 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2}
31 1+(0.786−0.618i)T2 1 + (0.786 - 0.618i)T^{2}
37 1+(−0.771+0.308i)T+(0.723−0.690i)T2 1 + (-0.771 + 0.308i)T + (0.723 - 0.690i)T^{2}
41 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
43 1+(−0.186−0.215i)T+(−0.142+0.989i)T2 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(1.11+1.56i)T+(−0.327+0.945i)T2 1 + (1.11 + 1.56i)T + (-0.327 + 0.945i)T^{2}
59 1+(−0.981−0.189i)T2 1 + (-0.981 - 0.189i)T^{2}
61 1+(−0.928−0.371i)T2 1 + (-0.928 - 0.371i)T^{2}
67 1+(−0.396+1.63i)T+(−0.888−0.458i)T2 1 + (-0.396 + 1.63i)T + (-0.888 - 0.458i)T^{2}
71 1+(−1.25+0.368i)T+(0.841−0.540i)T2 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2}
73 1+(0.995−0.0950i)T2 1 + (0.995 - 0.0950i)T^{2}
79 1+(1.11−1.56i)T+(−0.327−0.945i)T2 1 + (1.11 - 1.56i)T + (-0.327 - 0.945i)T^{2}
83 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
89 1+(0.786+0.618i)T2 1 + (0.786 + 0.618i)T^{2}
97 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{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

−10.06574158760829647814109334094, −9.515883982119087195449656645767, −8.038692432969917000884818847439, −7.64391010885453995342421941530, −6.80796357095822767583897871207, −6.04078620788856787493162367053, −5.13500158072987728661375727578, −4.36576810053629015871931720294, −3.61811680187849238465922337481, −2.17675129551046719322700988430, 1.48732436867437307256591231063, 2.60414407257608446010363697295, 3.53786837214564548243546473773, 4.42142220942085164110410386519, 5.29018356376492948981965467904, 5.96356660296916779920996676047, 7.23038461024177858962779639760, 8.109370271819049625454857603176, 9.298747153327053183920394347260, 10.18029974873476184292104996671

Graph of the ZZ-function along the critical line