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Properties

Label 2-2e7-128.101-c1-0-0
Degree 22
Conductor 128128
Sign −0.884−0.466i-0.884 - 0.466i
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  + (−1.29 − 0.567i)2-s + (−1.12 + 0.602i)3-s + (1.35 + 1.47i)4-s + (−1.29 − 1.58i)5-s + (1.80 − 0.140i)6-s + (−1.93 + 1.29i)7-s + (−0.922 − 2.67i)8-s + (−0.758 + 1.13i)9-s + (0.784 + 2.78i)10-s + (−4.58 + 1.39i)11-s + (−2.41 − 0.840i)12-s + (−1.77 − 1.45i)13-s + (3.24 − 0.577i)14-s + (2.41 + 1.00i)15-s + (−0.322 + 3.98i)16-s + (−0.698 + 0.289i)17-s + ⋯
L(s)  = 1  + (−0.915 − 0.401i)2-s + (−0.651 + 0.348i)3-s + (0.677 + 0.735i)4-s + (−0.580 − 0.707i)5-s + (0.736 − 0.0575i)6-s + (−0.732 + 0.489i)7-s + (−0.326 − 0.945i)8-s + (−0.252 + 0.378i)9-s + (0.248 + 0.881i)10-s + (−1.38 + 0.419i)11-s + (−0.697 − 0.242i)12-s + (−0.491 − 0.403i)13-s + (0.867 − 0.154i)14-s + (0.624 + 0.258i)15-s + (−0.0806 + 0.996i)16-s + (−0.169 + 0.0701i)17-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((−0.884−0.466i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+1/2) L(s)=((−0.884−0.466i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: −0.884−0.466i-0.884 - 0.466i
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), −0.884−0.466i)(2,\ 128,\ (\ :1/2),\ -0.884 - 0.466i)

Particular Values

L(1)L(1) ≈\approx 0.0200761+0.0811680i0.0200761 + 0.0811680i
L(12)L(\frac12) ≈\approx 0.0200761+0.0811680i0.0200761 + 0.0811680i
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+(1.29+0.567i)T 1 + (1.29 + 0.567i)T
good3 1+(1.12−0.602i)T+(1.66−2.49i)T2 1 + (1.12 - 0.602i)T + (1.66 - 2.49i)T^{2}
5 1+(1.29+1.58i)T+(−0.975+4.90i)T2 1 + (1.29 + 1.58i)T + (-0.975 + 4.90i)T^{2}
7 1+(1.93−1.29i)T+(2.67−6.46i)T2 1 + (1.93 - 1.29i)T + (2.67 - 6.46i)T^{2}
11 1+(4.58−1.39i)T+(9.14−6.11i)T2 1 + (4.58 - 1.39i)T + (9.14 - 6.11i)T^{2}
13 1+(1.77+1.45i)T+(2.53+12.7i)T2 1 + (1.77 + 1.45i)T + (2.53 + 12.7i)T^{2}
17 1+(0.698−0.289i)T+(12.0−12.0i)T2 1 + (0.698 - 0.289i)T + (12.0 - 12.0i)T^{2}
19 1+(−0.355−3.60i)T+(−18.6+3.70i)T2 1 + (-0.355 - 3.60i)T + (-18.6 + 3.70i)T^{2}
23 1+(0.824−0.164i)T+(21.2−8.80i)T2 1 + (0.824 - 0.164i)T + (21.2 - 8.80i)T^{2}
29 1+(−2.64+8.70i)T+(−24.1−16.1i)T2 1 + (-2.64 + 8.70i)T + (-24.1 - 16.1i)T^{2}
31 1+(−4.31−4.31i)T+31iT2 1 + (-4.31 - 4.31i)T + 31iT^{2}
37 1+(3.24+0.319i)T+(36.2+7.21i)T2 1 + (3.24 + 0.319i)T + (36.2 + 7.21i)T^{2}
41 1+(−2.34−11.7i)T+(−37.8+15.6i)T2 1 + (-2.34 - 11.7i)T + (-37.8 + 15.6i)T^{2}
43 1+(7.17+3.83i)T+(23.8+35.7i)T2 1 + (7.17 + 3.83i)T + (23.8 + 35.7i)T^{2}
47 1+(1.13+2.74i)T+(−33.2+33.2i)T2 1 + (1.13 + 2.74i)T + (-33.2 + 33.2i)T^{2}
53 1+(−2.85−9.39i)T+(−44.0+29.4i)T2 1 + (-2.85 - 9.39i)T + (-44.0 + 29.4i)T^{2}
59 1+(3.31−2.71i)T+(11.5−57.8i)T2 1 + (3.31 - 2.71i)T + (11.5 - 57.8i)T^{2}
61 1+(−0.0672−0.125i)T+(−33.8+50.7i)T2 1 + (-0.0672 - 0.125i)T + (-33.8 + 50.7i)T^{2}
67 1+(−3.12−5.84i)T+(−37.2+55.7i)T2 1 + (-3.12 - 5.84i)T + (-37.2 + 55.7i)T^{2}
71 1+(6.30+9.44i)T+(−27.1+65.5i)T2 1 + (6.30 + 9.44i)T + (-27.1 + 65.5i)T^{2}
73 1+(4.87+3.25i)T+(27.9+67.4i)T2 1 + (4.87 + 3.25i)T + (27.9 + 67.4i)T^{2}
79 1+(−4.87+11.7i)T+(−55.8−55.8i)T2 1 + (-4.87 + 11.7i)T + (-55.8 - 55.8i)T^{2}
83 1+(13.3−1.31i)T+(81.4−16.1i)T2 1 + (13.3 - 1.31i)T + (81.4 - 16.1i)T^{2}
89 1+(13.5+2.69i)T+(82.2+34.0i)T2 1 + (13.5 + 2.69i)T + (82.2 + 34.0i)T^{2}
97 1+(−4.07−4.07i)T+97iT2 1 + (-4.07 - 4.07i)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

−13.32335744074211760729144068043, −12.34840697983622234347436161768, −11.77842449631118821893728096149, −10.44190651148850590557739739689, −9.905348477908002946422277846618, −8.446749484721161513316955314748, −7.73834141239905790759388175941, −6.05117439073838575695675335182, −4.66578565724831013840941935545, −2.73116661081429475937941782681, 0.11664435912694360078336561450, 2.99566202736871662362920409262, 5.35519824839387940282491633753, 6.71183726849112582942047611756, 7.22076497679749874145794937309, 8.551522509600294001528306219581, 9.887065048400168638026721800232, 10.83943378153729385419970107557, 11.52844298484460107699807529436, 12.76343307490831403000124251007

Graph of the ZZ-function along the critical line