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Properties

Label 2-2e7-128.101-c1-0-10
Degree 22
Conductor 128128
Sign 0.981+0.189i0.981 + 0.189i
Analytic cond. 1.022081.02208
Root an. cond. 1.010981.01098
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.793 + 1.17i)2-s + (1.68 − 0.898i)3-s + (−0.742 − 1.85i)4-s + (−1.56 − 1.91i)5-s + (−0.281 + 2.68i)6-s + (3.63 − 2.43i)7-s + (2.76 + 0.603i)8-s + (0.353 − 0.529i)9-s + (3.48 − 0.321i)10-s + (−4.39 + 1.33i)11-s + (−2.91 − 2.45i)12-s + (1.74 + 1.42i)13-s + (−0.0388 + 6.19i)14-s + (−4.35 − 1.80i)15-s + (−2.89 + 2.75i)16-s + (3.75 − 1.55i)17-s + ⋯
L(s)  = 1  + (−0.560 + 0.827i)2-s + (0.970 − 0.518i)3-s + (−0.371 − 0.928i)4-s + (−0.701 − 0.854i)5-s + (−0.114 + 1.09i)6-s + (1.37 − 0.919i)7-s + (0.976 + 0.213i)8-s + (0.117 − 0.176i)9-s + (1.10 − 0.101i)10-s + (−1.32 + 0.401i)11-s + (−0.842 − 0.709i)12-s + (0.483 + 0.396i)13-s + (−0.0103 + 1.65i)14-s + (−1.12 − 0.466i)15-s + (−0.724 + 0.689i)16-s + (0.909 − 0.376i)17-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((0.981+0.189i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+1/2) L(s)=((0.981+0.189i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 0.981+0.189i0.981 + 0.189i
Analytic conductor: 1.022081.02208
Root analytic conductor: 1.010981.01098
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ128(101,⋅)\chi_{128} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 128, ( :1/2), 0.981+0.189i)(2,\ 128,\ (\ :1/2),\ 0.981 + 0.189i)

Particular Values

L(1)L(1) ≈\approx 1.02299−0.0980705i1.02299 - 0.0980705i
L(12)L(\frac12) ≈\approx 1.02299−0.0980705i1.02299 - 0.0980705i
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)
bad2 1+(0.793−1.17i)T 1 + (0.793 - 1.17i)T
good3 1+(−1.68+0.898i)T+(1.66−2.49i)T2 1 + (-1.68 + 0.898i)T + (1.66 - 2.49i)T^{2}
5 1+(1.56+1.91i)T+(−0.975+4.90i)T2 1 + (1.56 + 1.91i)T + (-0.975 + 4.90i)T^{2}
7 1+(−3.63+2.43i)T+(2.67−6.46i)T2 1 + (-3.63 + 2.43i)T + (2.67 - 6.46i)T^{2}
11 1+(4.39−1.33i)T+(9.14−6.11i)T2 1 + (4.39 - 1.33i)T + (9.14 - 6.11i)T^{2}
13 1+(−1.74−1.42i)T+(2.53+12.7i)T2 1 + (-1.74 - 1.42i)T + (2.53 + 12.7i)T^{2}
17 1+(−3.75+1.55i)T+(12.0−12.0i)T2 1 + (-3.75 + 1.55i)T + (12.0 - 12.0i)T^{2}
19 1+(−0.804−8.16i)T+(−18.6+3.70i)T2 1 + (-0.804 - 8.16i)T + (-18.6 + 3.70i)T^{2}
23 1+(1.74−0.347i)T+(21.2−8.80i)T2 1 + (1.74 - 0.347i)T + (21.2 - 8.80i)T^{2}
29 1+(−0.598+1.97i)T+(−24.1−16.1i)T2 1 + (-0.598 + 1.97i)T + (-24.1 - 16.1i)T^{2}
31 1+(3.81+3.81i)T+31iT2 1 + (3.81 + 3.81i)T + 31iT^{2}
37 1+(1.04+0.102i)T+(36.2+7.21i)T2 1 + (1.04 + 0.102i)T + (36.2 + 7.21i)T^{2}
41 1+(−0.711−3.57i)T+(−37.8+15.6i)T2 1 + (-0.711 - 3.57i)T + (-37.8 + 15.6i)T^{2}
43 1+(0.793+0.423i)T+(23.8+35.7i)T2 1 + (0.793 + 0.423i)T + (23.8 + 35.7i)T^{2}
47 1+(−1.43−3.46i)T+(−33.2+33.2i)T2 1 + (-1.43 - 3.46i)T + (-33.2 + 33.2i)T^{2}
53 1+(−1.79−5.91i)T+(−44.0+29.4i)T2 1 + (-1.79 - 5.91i)T + (-44.0 + 29.4i)T^{2}
59 1+(4.26−3.49i)T+(11.5−57.8i)T2 1 + (4.26 - 3.49i)T + (11.5 - 57.8i)T^{2}
61 1+(2.89+5.41i)T+(−33.8+50.7i)T2 1 + (2.89 + 5.41i)T + (-33.8 + 50.7i)T^{2}
67 1+(2.61+4.89i)T+(−37.2+55.7i)T2 1 + (2.61 + 4.89i)T + (-37.2 + 55.7i)T^{2}
71 1+(−0.916−1.37i)T+(−27.1+65.5i)T2 1 + (-0.916 - 1.37i)T + (-27.1 + 65.5i)T^{2}
73 1+(−6.52−4.36i)T+(27.9+67.4i)T2 1 + (-6.52 - 4.36i)T + (27.9 + 67.4i)T^{2}
79 1+(4.05−9.80i)T+(−55.8−55.8i)T2 1 + (4.05 - 9.80i)T + (-55.8 - 55.8i)T^{2}
83 1+(0.630−0.0621i)T+(81.4−16.1i)T2 1 + (0.630 - 0.0621i)T + (81.4 - 16.1i)T^{2}
89 1+(−3.26−0.649i)T+(82.2+34.0i)T2 1 + (-3.26 - 0.649i)T + (82.2 + 34.0i)T^{2}
97 1+(12.6+12.6i)T+97iT2 1 + (12.6 + 12.6i)T + 97iT^{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

−13.76355045996499708643853754561, −12.51954024286557126994115700370, −11.08739180895677751576310876865, −9.959551090229819163050929908092, −8.507370103133028687115291104304, −7.76869991972290422110601982939, −7.66199479188589657737530140411, −5.43741241230721573118911457962, −4.24003204819980336188134041832, −1.54360325812182180933004949612, 2.54110431857810722440460355564, 3.45210484051890464526484260178, 5.09211553610704047339388490087, 7.54490164628953767390668762033, 8.301739592640086702681419332310, 9.006578995458521407068255385024, 10.47816478386126404860251873630, 11.11111147183805602827016805586, 12.03027316090919291868381780690, 13.38296094840089922269989269265

Graph of the ZZ-function along the critical line