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Properties

Label 2-2e7-128.101-c3-0-23
Degree 22
Conductor 128128
Sign 0.124−0.992i0.124 - 0.992i
Analytic cond. 7.552247.55224
Root an. cond. 2.748132.74813
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.35 + 1.56i)2-s + (−0.400 + 0.213i)3-s + (3.10 + 7.37i)4-s + (4.33 + 5.27i)5-s + (−1.27 − 0.121i)6-s + (20.4 − 13.6i)7-s + (−4.20 + 22.2i)8-s + (−14.8 + 22.2i)9-s + (1.95 + 19.2i)10-s + (7.02 − 2.13i)11-s + (−2.82 − 2.28i)12-s + (0.932 + 0.765i)13-s + (69.4 − 0.220i)14-s + (−2.86 − 1.18i)15-s + (−44.6 + 45.8i)16-s + (−17.4 + 7.21i)17-s + ⋯
L(s)  = 1  + (0.833 + 0.552i)2-s + (−0.0770 + 0.0411i)3-s + (0.388 + 0.921i)4-s + (0.387 + 0.472i)5-s + (−0.0869 − 0.00828i)6-s + (1.10 − 0.736i)7-s + (−0.185 + 0.982i)8-s + (−0.551 + 0.825i)9-s + (0.0617 + 0.607i)10-s + (0.192 − 0.0583i)11-s + (−0.0678 − 0.0549i)12-s + (0.0198 + 0.0163i)13-s + (1.32 − 0.00421i)14-s + (−0.0492 − 0.0204i)15-s + (−0.698 + 0.716i)16-s + (−0.248 + 0.103i)17-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((0.124−0.992i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+3/2) L(s)=((0.124−0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 0.124−0.992i0.124 - 0.992i
Analytic conductor: 7.552247.55224
Root analytic conductor: 2.748132.74813
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ128(101,⋅)\chi_{128} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 128, ( :3/2), 0.124−0.992i)(2,\ 128,\ (\ :3/2),\ 0.124 - 0.992i)

Particular Values

L(2)L(2) ≈\approx 2.09170+1.84532i2.09170 + 1.84532i
L(12)L(\frac12) ≈\approx 2.09170+1.84532i2.09170 + 1.84532i
L(52)L(\frac{5}{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+(−2.35−1.56i)T 1 + (-2.35 - 1.56i)T
good3 1+(0.400−0.213i)T+(15.0−22.4i)T2 1 + (0.400 - 0.213i)T + (15.0 - 22.4i)T^{2}
5 1+(−4.33−5.27i)T+(−24.3+122.i)T2 1 + (-4.33 - 5.27i)T + (-24.3 + 122. i)T^{2}
7 1+(−20.4+13.6i)T+(131.−316.i)T2 1 + (-20.4 + 13.6i)T + (131. - 316. i)T^{2}
11 1+(−7.02+2.13i)T+(1.10e3−739.i)T2 1 + (-7.02 + 2.13i)T + (1.10e3 - 739. i)T^{2}
13 1+(−0.932−0.765i)T+(428.+2.15e3i)T2 1 + (-0.932 - 0.765i)T + (428. + 2.15e3i)T^{2}
17 1+(17.4−7.21i)T+(3.47e3−3.47e3i)T2 1 + (17.4 - 7.21i)T + (3.47e3 - 3.47e3i)T^{2}
19 1+(−10.2−103.i)T+(−6.72e3+1.33e3i)T2 1 + (-10.2 - 103. i)T + (-6.72e3 + 1.33e3i)T^{2}
23 1+(−23.9+4.76i)T+(1.12e4−4.65e3i)T2 1 + (-23.9 + 4.76i)T + (1.12e4 - 4.65e3i)T^{2}
29 1+(−83.8+276.i)T+(−2.02e4−1.35e4i)T2 1 + (-83.8 + 276. i)T + (-2.02e4 - 1.35e4i)T^{2}
31 1+(−12.7−12.7i)T+2.97e4iT2 1 + (-12.7 - 12.7i)T + 2.97e4iT^{2}
37 1+(−54.8−5.40i)T+(4.96e4+9.88e3i)T2 1 + (-54.8 - 5.40i)T + (4.96e4 + 9.88e3i)T^{2}
41 1+(82.2+413.i)T+(−6.36e4+2.63e4i)T2 1 + (82.2 + 413. i)T + (-6.36e4 + 2.63e4i)T^{2}
43 1+(−23.2−12.4i)T+(4.41e4+6.61e4i)T2 1 + (-23.2 - 12.4i)T + (4.41e4 + 6.61e4i)T^{2}
47 1+(137.+331.i)T+(−7.34e4+7.34e4i)T2 1 + (137. + 331. i)T + (-7.34e4 + 7.34e4i)T^{2}
53 1+(−108.−358.i)T+(−1.23e5+8.27e4i)T2 1 + (-108. - 358. i)T + (-1.23e5 + 8.27e4i)T^{2}
59 1+(−421.+345.i)T+(4.00e4−2.01e5i)T2 1 + (-421. + 345. i)T + (4.00e4 - 2.01e5i)T^{2}
61 1+(201.+376.i)T+(−1.26e5+1.88e5i)T2 1 + (201. + 376. i)T + (-1.26e5 + 1.88e5i)T^{2}
67 1+(157.+295.i)T+(−1.67e5+2.50e5i)T2 1 + (157. + 295. i)T + (-1.67e5 + 2.50e5i)T^{2}
71 1+(−275.−412.i)T+(−1.36e5+3.30e5i)T2 1 + (-275. - 412. i)T + (-1.36e5 + 3.30e5i)T^{2}
73 1+(−32.5−21.7i)T+(1.48e5+3.59e5i)T2 1 + (-32.5 - 21.7i)T + (1.48e5 + 3.59e5i)T^{2}
79 1+(146.−353.i)T+(−3.48e5−3.48e5i)T2 1 + (146. - 353. i)T + (-3.48e5 - 3.48e5i)T^{2}
83 1+(−894.+88.0i)T+(5.60e5−1.11e5i)T2 1 + (-894. + 88.0i)T + (5.60e5 - 1.11e5i)T^{2}
89 1+(1.06e3+211.i)T+(6.51e5+2.69e5i)T2 1 + (1.06e3 + 211. i)T + (6.51e5 + 2.69e5i)T^{2}
97 1+(16.0+16.0i)T+9.12e5iT2 1 + (16.0 + 16.0i)T + 9.12e5iT^{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

−13.56474591820841784068217869111, −12.09994564331443843171065193731, −11.17657409558357846525647524807, −10.30710594845612098509497058491, −8.382892211177131819864187593264, −7.65313279547126354846766876614, −6.34767495749388321145250930306, −5.20223049448466535548743169040, −4.01898231629231877090086691216, −2.23514827123164350155072920444, 1.32122017426703788322939835409, 2.91319100730070213813455960977, 4.70965709274382309483053398069, 5.53063319844657791439719592155, 6.79974193832863545619017161772, 8.667264324784947403059522524982, 9.460017200996961145390777261054, 11.02421891438437214605278674212, 11.65388038299567089673591936272, 12.57410947197526205078629839761

Graph of the ZZ-function along the critical line