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Properties

Label 2-2e7-128.101-c3-0-25
Degree 22
Conductor 128128
Sign −0.402+0.915i-0.402 + 0.915i
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  + (−0.130 − 2.82i)2-s + (−5.68 + 3.04i)3-s + (−7.96 + 0.735i)4-s + (4.50 + 5.49i)5-s + (9.32 + 15.6i)6-s + (19.5 − 13.0i)7-s + (3.11 + 22.4i)8-s + (8.10 − 12.1i)9-s + (14.9 − 13.4i)10-s + (6.71 − 2.03i)11-s + (43.0 − 28.4i)12-s + (−65.6 − 53.8i)13-s + (−39.3 − 53.4i)14-s + (−42.3 − 17.5i)15-s + (62.9 − 11.7i)16-s + (−7.92 + 3.28i)17-s + ⋯
L(s)  = 1  + (−0.0460 − 0.998i)2-s + (−1.09 + 0.585i)3-s + (−0.995 + 0.0919i)4-s + (0.403 + 0.491i)5-s + (0.634 + 1.06i)6-s + (1.05 − 0.704i)7-s + (0.137 + 0.990i)8-s + (0.300 − 0.449i)9-s + (0.472 − 0.425i)10-s + (0.184 − 0.0558i)11-s + (1.03 − 0.683i)12-s + (−1.40 − 1.14i)13-s + (−0.751 − 1.02i)14-s + (−0.728 − 0.301i)15-s + (0.983 − 0.183i)16-s + (−0.113 + 0.0468i)17-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((−0.402+0.915i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+3/2) L(s)=((−0.402+0.915i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: −0.402+0.915i-0.402 + 0.915i
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), −0.402+0.915i)(2,\ 128,\ (\ :3/2),\ -0.402 + 0.915i)

Particular Values

L(2)L(2) ≈\approx 0.493265−0.755634i0.493265 - 0.755634i
L(12)L(\frac12) ≈\approx 0.493265−0.755634i0.493265 - 0.755634i
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+(0.130+2.82i)T 1 + (0.130 + 2.82i)T
good3 1+(5.68−3.04i)T+(15.0−22.4i)T2 1 + (5.68 - 3.04i)T + (15.0 - 22.4i)T^{2}
5 1+(−4.50−5.49i)T+(−24.3+122.i)T2 1 + (-4.50 - 5.49i)T + (-24.3 + 122. i)T^{2}
7 1+(−19.5+13.0i)T+(131.−316.i)T2 1 + (-19.5 + 13.0i)T + (131. - 316. i)T^{2}
11 1+(−6.71+2.03i)T+(1.10e3−739.i)T2 1 + (-6.71 + 2.03i)T + (1.10e3 - 739. i)T^{2}
13 1+(65.6+53.8i)T+(428.+2.15e3i)T2 1 + (65.6 + 53.8i)T + (428. + 2.15e3i)T^{2}
17 1+(7.92−3.28i)T+(3.47e3−3.47e3i)T2 1 + (7.92 - 3.28i)T + (3.47e3 - 3.47e3i)T^{2}
19 1+(14.1+143.i)T+(−6.72e3+1.33e3i)T2 1 + (14.1 + 143. i)T + (-6.72e3 + 1.33e3i)T^{2}
23 1+(−167.+33.2i)T+(1.12e4−4.65e3i)T2 1 + (-167. + 33.2i)T + (1.12e4 - 4.65e3i)T^{2}
29 1+(−11.5+38.1i)T+(−2.02e4−1.35e4i)T2 1 + (-11.5 + 38.1i)T + (-2.02e4 - 1.35e4i)T^{2}
31 1+(−218.−218.i)T+2.97e4iT2 1 + (-218. - 218. i)T + 2.97e4iT^{2}
37 1+(116.+11.4i)T+(4.96e4+9.88e3i)T2 1 + (116. + 11.4i)T + (4.96e4 + 9.88e3i)T^{2}
41 1+(29.9+150.i)T+(−6.36e4+2.63e4i)T2 1 + (29.9 + 150. i)T + (-6.36e4 + 2.63e4i)T^{2}
43 1+(162.+87.0i)T+(4.41e4+6.61e4i)T2 1 + (162. + 87.0i)T + (4.41e4 + 6.61e4i)T^{2}
47 1+(108.+261.i)T+(−7.34e4+7.34e4i)T2 1 + (108. + 261. i)T + (-7.34e4 + 7.34e4i)T^{2}
53 1+(−31.3−103.i)T+(−1.23e5+8.27e4i)T2 1 + (-31.3 - 103. i)T + (-1.23e5 + 8.27e4i)T^{2}
59 1+(−317.+260.i)T+(4.00e4−2.01e5i)T2 1 + (-317. + 260. i)T + (4.00e4 - 2.01e5i)T^{2}
61 1+(−30.9−57.8i)T+(−1.26e5+1.88e5i)T2 1 + (-30.9 - 57.8i)T + (-1.26e5 + 1.88e5i)T^{2}
67 1+(−364.−681.i)T+(−1.67e5+2.50e5i)T2 1 + (-364. - 681. i)T + (-1.67e5 + 2.50e5i)T^{2}
71 1+(607.+908.i)T+(−1.36e5+3.30e5i)T2 1 + (607. + 908. i)T + (-1.36e5 + 3.30e5i)T^{2}
73 1+(82.2+54.9i)T+(1.48e5+3.59e5i)T2 1 + (82.2 + 54.9i)T + (1.48e5 + 3.59e5i)T^{2}
79 1+(331.−800.i)T+(−3.48e5−3.48e5i)T2 1 + (331. - 800. i)T + (-3.48e5 - 3.48e5i)T^{2}
83 1+(218.−21.5i)T+(5.60e5−1.11e5i)T2 1 + (218. - 21.5i)T + (5.60e5 - 1.11e5i)T^{2}
89 1+(−486.−96.8i)T+(6.51e5+2.69e5i)T2 1 + (-486. - 96.8i)T + (6.51e5 + 2.69e5i)T^{2}
97 1+(209.+209.i)T+9.12e5iT2 1 + (209. + 209. i)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

−12.20335510804787863652943327328, −11.25776093140727976236385905916, −10.59447119674927495713736920444, −10.03047743018091294623641641948, −8.522338087286810531526597604367, −7.00469061207825127029308109585, −5.14451435432591272683941152680, −4.68744601416147563157888919026, −2.70548939856089165243913677414, −0.59665839960304718608044378793, 1.46181622408381153603136618539, 4.71004607571689383618695633503, 5.44032189814419927475013671832, 6.49705439582249615854736444593, 7.59233105653698472178379869874, 8.801553478629059663759766155670, 9.823717643254167182074195025583, 11.48284711121774353606538005387, 12.18880513770987105176571849156, 13.12660014886738142133438700124

Graph of the ZZ-function along the critical line