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Properties

Label 2-2e7-128.117-c1-0-6
Degree 22
Conductor 128128
Sign 0.909+0.416i0.909 + 0.416i
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  + (−0.766 + 1.18i)2-s + (−1.39 + 0.137i)3-s + (−0.823 − 1.82i)4-s + (1.67 − 3.12i)5-s + (0.906 − 1.76i)6-s + (0.194 − 0.979i)7-s + (2.79 + 0.419i)8-s + (−1.01 + 0.201i)9-s + (2.43 + 4.38i)10-s + (0.362 − 0.297i)11-s + (1.39 + 2.42i)12-s + (5.08 − 2.71i)13-s + (1.01 + 0.982i)14-s + (−1.90 + 4.59i)15-s + (−2.64 + 3.00i)16-s + (−0.701 − 1.69i)17-s + ⋯
L(s)  = 1  + (−0.542 + 0.840i)2-s + (−0.805 + 0.0793i)3-s + (−0.411 − 0.911i)4-s + (0.747 − 1.39i)5-s + (0.370 − 0.719i)6-s + (0.0736 − 0.370i)7-s + (0.988 + 0.148i)8-s + (−0.338 + 0.0672i)9-s + (0.769 + 1.38i)10-s + (0.109 − 0.0896i)11-s + (0.403 + 0.701i)12-s + (1.40 − 0.753i)13-s + (0.271 + 0.262i)14-s + (−0.491 + 1.18i)15-s + (−0.660 + 0.750i)16-s + (−0.170 − 0.410i)17-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((0.909+0.416i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+1/2) L(s)=((0.909+0.416i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 0.909+0.416i0.909 + 0.416i
Analytic conductor: 1.022081.02208
Root analytic conductor: 1.010981.01098
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ128(117,⋅)\chi_{128} (117, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 128, ( :1/2), 0.909+0.416i)(2,\ 128,\ (\ :1/2),\ 0.909 + 0.416i)

Particular Values

L(1)L(1) ≈\approx 0.660579−0.143966i0.660579 - 0.143966i
L(12)L(\frac12) ≈\approx 0.660579−0.143966i0.660579 - 0.143966i
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+(0.766−1.18i)T 1 + (0.766 - 1.18i)T
good3 1+(1.39−0.137i)T+(2.94−0.585i)T2 1 + (1.39 - 0.137i)T + (2.94 - 0.585i)T^{2}
5 1+(−1.67+3.12i)T+(−2.77−4.15i)T2 1 + (-1.67 + 3.12i)T + (-2.77 - 4.15i)T^{2}
7 1+(−0.194+0.979i)T+(−6.46−2.67i)T2 1 + (-0.194 + 0.979i)T + (-6.46 - 2.67i)T^{2}
11 1+(−0.362+0.297i)T+(2.14−10.7i)T2 1 + (-0.362 + 0.297i)T + (2.14 - 10.7i)T^{2}
13 1+(−5.08+2.71i)T+(7.22−10.8i)T2 1 + (-5.08 + 2.71i)T + (7.22 - 10.8i)T^{2}
17 1+(0.701+1.69i)T+(−12.0+12.0i)T2 1 + (0.701 + 1.69i)T + (-12.0 + 12.0i)T^{2}
19 1+(−0.364−0.110i)T+(15.7+10.5i)T2 1 + (-0.364 - 0.110i)T + (15.7 + 10.5i)T^{2}
23 1+(6.09+4.07i)T+(8.80+21.2i)T2 1 + (6.09 + 4.07i)T + (8.80 + 21.2i)T^{2}
29 1+(2.16−2.64i)T+(−5.65−28.4i)T2 1 + (2.16 - 2.64i)T + (-5.65 - 28.4i)T^{2}
31 1+(−6.95−6.95i)T+31iT2 1 + (-6.95 - 6.95i)T + 31iT^{2}
37 1+(0.132+0.436i)T+(−30.7+20.5i)T2 1 + (0.132 + 0.436i)T + (-30.7 + 20.5i)T^{2}
41 1+(3.08−4.62i)T+(−15.6−37.8i)T2 1 + (3.08 - 4.62i)T + (-15.6 - 37.8i)T^{2}
43 1+(−2.80−0.276i)T+(42.1+8.38i)T2 1 + (-2.80 - 0.276i)T + (42.1 + 8.38i)T^{2}
47 1+(8.00−3.31i)T+(33.2−33.2i)T2 1 + (8.00 - 3.31i)T + (33.2 - 33.2i)T^{2}
53 1+(−8.24−10.0i)T+(−10.3+51.9i)T2 1 + (-8.24 - 10.0i)T + (-10.3 + 51.9i)T^{2}
59 1+(5.72+3.05i)T+(32.7+49.0i)T2 1 + (5.72 + 3.05i)T + (32.7 + 49.0i)T^{2}
61 1+(−0.179−1.81i)T+(−59.8+11.9i)T2 1 + (-0.179 - 1.81i)T + (-59.8 + 11.9i)T^{2}
67 1+(0.243+2.47i)T+(−65.7+13.0i)T2 1 + (0.243 + 2.47i)T + (-65.7 + 13.0i)T^{2}
71 1+(−14.3−2.85i)T+(65.5+27.1i)T2 1 + (-14.3 - 2.85i)T + (65.5 + 27.1i)T^{2}
73 1+(−1.12−5.63i)T+(−67.4+27.9i)T2 1 + (-1.12 - 5.63i)T + (-67.4 + 27.9i)T^{2}
79 1+(1.06+0.443i)T+(55.8+55.8i)T2 1 + (1.06 + 0.443i)T + (55.8 + 55.8i)T^{2}
83 1+(−4.47+14.7i)T+(−69.0−46.1i)T2 1 + (-4.47 + 14.7i)T + (-69.0 - 46.1i)T^{2}
89 1+(11.2−7.52i)T+(34.0−82.2i)T2 1 + (11.2 - 7.52i)T + (34.0 - 82.2i)T^{2}
97 1+(−1.53−1.53i)T+97iT2 1 + (-1.53 - 1.53i)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

−13.48213717073942826380403602577, −12.33808725449340143361851193676, −10.96047560203678615852904285168, −10.06787992387036214617396140505, −8.830352040574722745228298775385, −8.182844948153348973255150808244, −6.37215820074939902874984749232, −5.62151296893654361400406332720, −4.63211809988441646876116492907, −1.03130998518253711823781677430, 2.12981158143748007574484360709, 3.71449897480664657987646458330, 5.83267561825445283796565631127, 6.70093391890964898094608878120, 8.278925615580897697629443247332, 9.560466092037535343089812522257, 10.49924738695237492359419916064, 11.35713002524675823483361444620, 11.85122053247400249788329795231, 13.41375608575706204420517441284

Graph of the ZZ-function along the critical line