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Properties

Label 2-1127-161.31-c0-0-0
Degree 22
Conductor 11271127
Sign 0.638−0.769i0.638 - 0.769i
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  + (−1.78 + 0.713i)2-s + (1.94 − 1.85i)4-s + (−1.34 + 2.93i)8-s + (0.981 + 0.189i)9-s + (0.771 + 0.308i)11-s + (0.167 − 3.50i)16-s + (−1.88 + 0.363i)18-s − 1.59·22-s + (−0.327 + 0.945i)23-s + (−0.786 − 0.618i)25-s + (0.273 − 0.0801i)29-s + (1.14 + 3.32i)32-s + (2.25 − 1.45i)36-s + (1.65 + 0.318i)37-s + (−0.544 − 1.19i)43-s + (2.06 − 0.828i)44-s + ⋯
L(s)  = 1  + (−1.78 + 0.713i)2-s + (1.94 − 1.85i)4-s + (−1.34 + 2.93i)8-s + (0.981 + 0.189i)9-s + (0.771 + 0.308i)11-s + (0.167 − 3.50i)16-s + (−1.88 + 0.363i)18-s − 1.59·22-s + (−0.327 + 0.945i)23-s + (−0.786 − 0.618i)25-s + (0.273 − 0.0801i)29-s + (1.14 + 3.32i)32-s + (2.25 − 1.45i)36-s + (1.65 + 0.318i)37-s + (−0.544 − 1.19i)43-s + (2.06 − 0.828i)44-s + ⋯

Functional equation

Λ(s)=(1127s/2 ΓC(s) L(s)=((0.638−0.769i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1127s/2 ΓC(s) L(s)=((0.638−0.769i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11271127    =    72⋅237^{2} \cdot 23
Sign: 0.638−0.769i0.638 - 0.769i
Analytic conductor: 0.5624460.562446
Root analytic conductor: 0.7499640.749964
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1127(31,⋅)\chi_{1127} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1127, ( :0), 0.638−0.769i)(2,\ 1127,\ (\ :0),\ 0.638 - 0.769i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.52402523710.5240252371
L(12)L(\frac12) ≈\approx 0.52402523710.5240252371
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+(0.327−0.945i)T 1 + (0.327 - 0.945i)T
good2 1+(1.78−0.713i)T+(0.723−0.690i)T2 1 + (1.78 - 0.713i)T + (0.723 - 0.690i)T^{2}
3 1+(−0.981−0.189i)T2 1 + (-0.981 - 0.189i)T^{2}
5 1+(0.786+0.618i)T2 1 + (0.786 + 0.618i)T^{2}
11 1+(−0.771−0.308i)T+(0.723+0.690i)T2 1 + (-0.771 - 0.308i)T + (0.723 + 0.690i)T^{2}
13 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
17 1+(0.888+0.458i)T2 1 + (0.888 + 0.458i)T^{2}
19 1+(0.888−0.458i)T2 1 + (0.888 - 0.458i)T^{2}
29 1+(−0.273+0.0801i)T+(0.841−0.540i)T2 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2}
31 1+(0.327−0.945i)T2 1 + (0.327 - 0.945i)T^{2}
37 1+(−1.65−0.318i)T+(0.928+0.371i)T2 1 + (-1.65 - 0.318i)T + (0.928 + 0.371i)T^{2}
41 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)T^{2}
43 1+(0.544+1.19i)T+(−0.654+0.755i)T2 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(−0.252+0.130i)T+(0.580−0.814i)T2 1 + (-0.252 + 0.130i)T + (0.580 - 0.814i)T^{2}
59 1+(0.995−0.0950i)T2 1 + (0.995 - 0.0950i)T^{2}
61 1+(−0.981+0.189i)T2 1 + (-0.981 + 0.189i)T^{2}
67 1+(−1.50−1.18i)T+(0.235+0.971i)T2 1 + (-1.50 - 1.18i)T + (0.235 + 0.971i)T^{2}
71 1+(0.118−0.822i)T+(−0.959−0.281i)T2 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2}
73 1+(−0.0475+0.998i)T2 1 + (-0.0475 + 0.998i)T^{2}
79 1+(−0.252−0.130i)T+(0.580+0.814i)T2 1 + (-0.252 - 0.130i)T + (0.580 + 0.814i)T^{2}
83 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
89 1+(0.327+0.945i)T2 1 + (0.327 + 0.945i)T^{2}
97 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)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

−9.804055547268963343124224065990, −9.457089058638790907327794120595, −8.435588739579030824801620468890, −7.76144178148624569814708276042, −7.02347794785143534536035596263, −6.37991516445185632423747939096, −5.41048462070235961124002868466, −4.07060767481808551899350581364, −2.25722802809866430814011184894, −1.23600636111698756311311627262, 1.06338571190988432946504560913, 2.13503069472381162042096921836, 3.36140842843713122473140111641, 4.32406654422176469176113533216, 6.19053536935441266448918712079, 6.87026208858937101552438070450, 7.75285274015612189776583228334, 8.377404708513062911600608201723, 9.432223485688093580567907747646, 9.635422470322342298643234316209

Graph of the ZZ-function along the critical line