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Properties

Label 2-2e7-128.117-c1-0-1
Degree 22
Conductor 128128
Sign −0.996+0.0861i-0.996 + 0.0861i
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  + (−0.0717 + 1.41i)2-s + (−1.31 + 0.129i)3-s + (−1.98 − 0.202i)4-s + (−1.48 + 2.78i)5-s + (−0.0886 − 1.87i)6-s + (0.373 − 1.87i)7-s + (0.429 − 2.79i)8-s + (−1.22 + 0.243i)9-s + (−3.82 − 2.30i)10-s + (−2.23 + 1.83i)11-s + (2.64 + 0.00899i)12-s + (−2.26 + 1.21i)13-s + (2.62 + 0.662i)14-s + (1.59 − 3.86i)15-s + (3.91 + 0.806i)16-s + (2.75 + 6.64i)17-s + ⋯
L(s)  = 1  + (−0.0507 + 0.998i)2-s + (−0.760 + 0.0749i)3-s + (−0.994 − 0.101i)4-s + (−0.665 + 1.24i)5-s + (−0.0362 − 0.763i)6-s + (0.141 − 0.710i)7-s + (0.151 − 0.988i)8-s + (−0.407 + 0.0810i)9-s + (−1.20 − 0.727i)10-s + (−0.673 + 0.552i)11-s + (0.764 + 0.00259i)12-s + (−0.628 + 0.335i)13-s + (0.702 + 0.177i)14-s + (0.413 − 0.997i)15-s + (0.979 + 0.201i)16-s + (0.667 + 1.61i)17-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((−0.996+0.0861i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0861i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+1/2) L(s)=((−0.996+0.0861i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0861i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: −0.996+0.0861i-0.996 + 0.0861i
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), −0.996+0.0861i)(2,\ 128,\ (\ :1/2),\ -0.996 + 0.0861i)

Particular Values

L(1)L(1) ≈\approx 0.0199412−0.461905i0.0199412 - 0.461905i
L(12)L(\frac12) ≈\approx 0.0199412−0.461905i0.0199412 - 0.461905i
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+(0.0717−1.41i)T 1 + (0.0717 - 1.41i)T
good3 1+(1.31−0.129i)T+(2.94−0.585i)T2 1 + (1.31 - 0.129i)T + (2.94 - 0.585i)T^{2}
5 1+(1.48−2.78i)T+(−2.77−4.15i)T2 1 + (1.48 - 2.78i)T + (-2.77 - 4.15i)T^{2}
7 1+(−0.373+1.87i)T+(−6.46−2.67i)T2 1 + (-0.373 + 1.87i)T + (-6.46 - 2.67i)T^{2}
11 1+(2.23−1.83i)T+(2.14−10.7i)T2 1 + (2.23 - 1.83i)T + (2.14 - 10.7i)T^{2}
13 1+(2.26−1.21i)T+(7.22−10.8i)T2 1 + (2.26 - 1.21i)T + (7.22 - 10.8i)T^{2}
17 1+(−2.75−6.64i)T+(−12.0+12.0i)T2 1 + (-2.75 - 6.64i)T + (-12.0 + 12.0i)T^{2}
19 1+(−4.33−1.31i)T+(15.7+10.5i)T2 1 + (-4.33 - 1.31i)T + (15.7 + 10.5i)T^{2}
23 1+(−0.470−0.314i)T+(8.80+21.2i)T2 1 + (-0.470 - 0.314i)T + (8.80 + 21.2i)T^{2}
29 1+(0.777−0.947i)T+(−5.65−28.4i)T2 1 + (0.777 - 0.947i)T + (-5.65 - 28.4i)T^{2}
31 1+(3.28+3.28i)T+31iT2 1 + (3.28 + 3.28i)T + 31iT^{2}
37 1+(−2.08−6.85i)T+(−30.7+20.5i)T2 1 + (-2.08 - 6.85i)T + (-30.7 + 20.5i)T^{2}
41 1+(−4.65+6.96i)T+(−15.6−37.8i)T2 1 + (-4.65 + 6.96i)T + (-15.6 - 37.8i)T^{2}
43 1+(8.37+0.824i)T+(42.1+8.38i)T2 1 + (8.37 + 0.824i)T + (42.1 + 8.38i)T^{2}
47 1+(11.2−4.67i)T+(33.2−33.2i)T2 1 + (11.2 - 4.67i)T + (33.2 - 33.2i)T^{2}
53 1+(2.90+3.54i)T+(−10.3+51.9i)T2 1 + (2.90 + 3.54i)T + (-10.3 + 51.9i)T^{2}
59 1+(−8.29−4.43i)T+(32.7+49.0i)T2 1 + (-8.29 - 4.43i)T + (32.7 + 49.0i)T^{2}
61 1+(0.811+8.23i)T+(−59.8+11.9i)T2 1 + (0.811 + 8.23i)T + (-59.8 + 11.9i)T^{2}
67 1+(0.215+2.19i)T+(−65.7+13.0i)T2 1 + (0.215 + 2.19i)T + (-65.7 + 13.0i)T^{2}
71 1+(−6.18−1.22i)T+(65.5+27.1i)T2 1 + (-6.18 - 1.22i)T + (65.5 + 27.1i)T^{2}
73 1+(−2.35−11.8i)T+(−67.4+27.9i)T2 1 + (-2.35 - 11.8i)T + (-67.4 + 27.9i)T^{2}
79 1+(2.74+1.13i)T+(55.8+55.8i)T2 1 + (2.74 + 1.13i)T + (55.8 + 55.8i)T^{2}
83 1+(−1.28+4.23i)T+(−69.0−46.1i)T2 1 + (-1.28 + 4.23i)T + (-69.0 - 46.1i)T^{2}
89 1+(−12.6+8.44i)T+(34.0−82.2i)T2 1 + (-12.6 + 8.44i)T + (34.0 - 82.2i)T^{2}
97 1+(−6.54−6.54i)T+97iT2 1 + (-6.54 - 6.54i)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

−14.32457474949830252480263778983, −12.95907964535700657257577739606, −11.73343769711173988966374788376, −10.65387568730581470624189832976, −9.930150138926385217985657854493, −8.060226375475790552903911321582, −7.34517409175370694173240413464, −6.31937655731691024337732980774, −5.07560159815284717773378902561, −3.61515596763959349715060226890, 0.54774668922782752108278553158, 3.01683119381176204933572057881, 5.02042448383212418728705194454, 5.36030841117322579456432740497, 7.76399101244531153734752801244, 8.764985745810461328147468518644, 9.697570532543029423220210264062, 11.19766358345169616167616918683, 11.82259628832744169848846541781, 12.41646129155511882977057194811

Graph of the ZZ-function along the critical line