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Properties

Label 2-2e7-128.117-c1-0-8
Degree 22
Conductor 128128
Sign 0.917−0.398i0.917 - 0.398i
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  + (1.37 + 0.321i)2-s + (−0.194 + 0.0191i)3-s + (1.79 + 0.884i)4-s + (−0.448 + 0.838i)5-s + (−0.274 − 0.0361i)6-s + (0.394 − 1.98i)7-s + (2.18 + 1.79i)8-s + (−2.90 + 0.577i)9-s + (−0.886 + 1.01i)10-s + (1.96 − 1.61i)11-s + (−0.366 − 0.137i)12-s + (−3.43 + 1.83i)13-s + (1.18 − 2.60i)14-s + (0.0712 − 0.172i)15-s + (2.43 + 3.17i)16-s + (−2.58 − 6.25i)17-s + ⋯
L(s)  = 1  + (0.973 + 0.227i)2-s + (−0.112 + 0.0110i)3-s + (0.896 + 0.442i)4-s + (−0.200 + 0.375i)5-s + (−0.112 − 0.0147i)6-s + (0.149 − 0.749i)7-s + (0.773 + 0.634i)8-s + (−0.968 + 0.192i)9-s + (−0.280 + 0.319i)10-s + (0.591 − 0.485i)11-s + (−0.105 − 0.0398i)12-s + (−0.952 + 0.509i)13-s + (0.315 − 0.696i)14-s + (0.0184 − 0.0444i)15-s + (0.608 + 0.793i)16-s + (−0.627 − 1.51i)17-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((0.917−0.398i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+1/2) L(s)=((0.917−0.398i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 0.917−0.398i0.917 - 0.398i
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), 0.917−0.398i)(2,\ 128,\ (\ :1/2),\ 0.917 - 0.398i)

Particular Values

L(1)L(1) ≈\approx 1.62901+0.338223i1.62901 + 0.338223i
L(12)L(\frac12) ≈\approx 1.62901+0.338223i1.62901 + 0.338223i
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+(−1.37−0.321i)T 1 + (-1.37 - 0.321i)T
good3 1+(0.194−0.0191i)T+(2.94−0.585i)T2 1 + (0.194 - 0.0191i)T + (2.94 - 0.585i)T^{2}
5 1+(0.448−0.838i)T+(−2.77−4.15i)T2 1 + (0.448 - 0.838i)T + (-2.77 - 4.15i)T^{2}
7 1+(−0.394+1.98i)T+(−6.46−2.67i)T2 1 + (-0.394 + 1.98i)T + (-6.46 - 2.67i)T^{2}
11 1+(−1.96+1.61i)T+(2.14−10.7i)T2 1 + (-1.96 + 1.61i)T + (2.14 - 10.7i)T^{2}
13 1+(3.43−1.83i)T+(7.22−10.8i)T2 1 + (3.43 - 1.83i)T + (7.22 - 10.8i)T^{2}
17 1+(2.58+6.25i)T+(−12.0+12.0i)T2 1 + (2.58 + 6.25i)T + (-12.0 + 12.0i)T^{2}
19 1+(−1.58−0.481i)T+(15.7+10.5i)T2 1 + (-1.58 - 0.481i)T + (15.7 + 10.5i)T^{2}
23 1+(4.28+2.86i)T+(8.80+21.2i)T2 1 + (4.28 + 2.86i)T + (8.80 + 21.2i)T^{2}
29 1+(3.65−4.45i)T+(−5.65−28.4i)T2 1 + (3.65 - 4.45i)T + (-5.65 - 28.4i)T^{2}
31 1+(1.49+1.49i)T+31iT2 1 + (1.49 + 1.49i)T + 31iT^{2}
37 1+(−0.443−1.46i)T+(−30.7+20.5i)T2 1 + (-0.443 - 1.46i)T + (-30.7 + 20.5i)T^{2}
41 1+(−1.49+2.23i)T+(−15.6−37.8i)T2 1 + (-1.49 + 2.23i)T + (-15.6 - 37.8i)T^{2}
43 1+(−10.6−1.05i)T+(42.1+8.38i)T2 1 + (-10.6 - 1.05i)T + (42.1 + 8.38i)T^{2}
47 1+(1.16−0.483i)T+(33.2−33.2i)T2 1 + (1.16 - 0.483i)T + (33.2 - 33.2i)T^{2}
53 1+(−5.69−6.93i)T+(−10.3+51.9i)T2 1 + (-5.69 - 6.93i)T + (-10.3 + 51.9i)T^{2}
59 1+(1.42+0.761i)T+(32.7+49.0i)T2 1 + (1.42 + 0.761i)T + (32.7 + 49.0i)T^{2}
61 1+(−0.332−3.37i)T+(−59.8+11.9i)T2 1 + (-0.332 - 3.37i)T + (-59.8 + 11.9i)T^{2}
67 1+(0.689+6.99i)T+(−65.7+13.0i)T2 1 + (0.689 + 6.99i)T + (-65.7 + 13.0i)T^{2}
71 1+(7.67+1.52i)T+(65.5+27.1i)T2 1 + (7.67 + 1.52i)T + (65.5 + 27.1i)T^{2}
73 1+(−0.201−1.01i)T+(−67.4+27.9i)T2 1 + (-0.201 - 1.01i)T + (-67.4 + 27.9i)T^{2}
79 1+(−13.7−5.68i)T+(55.8+55.8i)T2 1 + (-13.7 - 5.68i)T + (55.8 + 55.8i)T^{2}
83 1+(4.86−16.0i)T+(−69.0−46.1i)T2 1 + (4.86 - 16.0i)T + (-69.0 - 46.1i)T^{2}
89 1+(−3.43+2.29i)T+(34.0−82.2i)T2 1 + (-3.43 + 2.29i)T + (34.0 - 82.2i)T^{2}
97 1+(2.61+2.61i)T+97iT2 1 + (2.61 + 2.61i)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

−13.88652973577278203056725529470, −12.34734421677500706895817564684, −11.44858832526688242639254044723, −10.81356096432277234563760535091, −9.157673383030713216975799955780, −7.61723990626742097614671714435, −6.81968061437001885752769531783, −5.46502652224407135104389729903, −4.20437919970744213361785557843, −2.74807047629660629955399762809, 2.31774303233298479256069002590, 4.00648795899222448798296577920, 5.35599550690896296346651729793, 6.26876987761105405330152098634, 7.80664412898919681221695353818, 9.113892659434563214486975913960, 10.43516904324412324326654474066, 11.66857183307663438416410174160, 12.20162109319998872676656227536, 13.09320794422319658183193402657

Graph of the ZZ-function along the critical line