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Properties

Label 2-1127-161.114-c0-0-10
Degree 22
Conductor 11271127
Sign −0.198−0.980i-0.198 - 0.980i
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  + (0.866 − 1.5i)2-s + (−0.965 − 1.67i)3-s + (−1 − 1.73i)4-s − 3.34·6-s − 1.73·8-s + (−1.36 + 2.36i)9-s + (−1.93 + 3.34i)12-s − 0.517·13-s + (−0.5 + 0.866i)16-s + (2.36 + 4.09i)18-s + (0.5 − 0.866i)23-s + (1.67 + 2.89i)24-s + (−0.5 − 0.866i)25-s + (−0.448 + 0.776i)26-s + 3.34·27-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)2-s + (−0.965 − 1.67i)3-s + (−1 − 1.73i)4-s − 3.34·6-s − 1.73·8-s + (−1.36 + 2.36i)9-s + (−1.93 + 3.34i)12-s − 0.517·13-s + (−0.5 + 0.866i)16-s + (2.36 + 4.09i)18-s + (0.5 − 0.866i)23-s + (1.67 + 2.89i)24-s + (−0.5 − 0.866i)25-s + (−0.448 + 0.776i)26-s + 3.34·27-s + ⋯

Functional equation

Λ(s)=(1127s/2 ΓC(s) L(s)=((−0.198−0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1127s/2 ΓC(s) L(s)=((−0.198−0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11271127    =    72⋅237^{2} \cdot 23
Sign: −0.198−0.980i-0.198 - 0.980i
Analytic conductor: 0.5624460.562446
Root analytic conductor: 0.7499640.749964
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1127(275,⋅)\chi_{1127} (275, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1127, ( :0), −0.198−0.980i)(2,\ 1127,\ (\ :0),\ -0.198 - 0.980i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.92852983700.9285298370
L(12)L(\frac12) ≈\approx 0.92852983700.9285298370
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good2 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
3 1+(0.965+1.67i)T+(−0.5+0.866i)T2 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+0.517T+T2 1 + 0.517T + T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+T+T2 1 + T + T^{2}
31 1+(0.258+0.448i)T+(−0.5+0.866i)T2 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1−0.517T+T2 1 - 0.517T + T^{2}
43 1−T2 1 - T^{2}
47 1+(−0.965+1.67i)T+(−0.5−0.866i)T2 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(−0.707−1.22i)T+(−0.5+0.866i)T2 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−T+T2 1 - T + T^{2}
73 1+(−0.965−1.67i)T+(−0.5+0.866i)T2 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1−T2 1 - 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

−9.894542718334242637906424253592, −8.629896176126406306329784198465, −7.57646521948628122433590953866, −6.76155701399814990455880096243, −5.77336175884207078222558070825, −5.19740918896636854673061045664, −4.11293076808370701215598876342, −2.60779010794368938140809883660, −1.99753348293486890208556830139, −0.71562465124332083766634630029, 3.33043958185879804505894440870, 4.01552716753520345773326983628, 4.90563425908324048444096631265, 5.42652738038804516429644147245, 6.08111969176445465990155292687, 7.01217238402317070245955272983, 7.927188772760102547720021296851, 9.179752039445669890540708111957, 9.491105003187185479668862968098, 10.67172955098877993140118682886

Graph of the ZZ-function along the critical line