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Properties

Label 2-2e7-128.101-c3-0-22
Degree 22
Conductor 128128
Sign 0.836+0.547i0.836 + 0.547i
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  + (−0.567 + 2.77i)2-s + (−5.84 + 3.12i)3-s + (−7.35 − 3.14i)4-s + (0.807 + 0.983i)5-s + (−5.33 − 17.9i)6-s + (−28.4 + 18.9i)7-s + (12.8 − 18.5i)8-s + (9.36 − 14.0i)9-s + (−3.18 + 1.67i)10-s + (45.4 − 13.7i)11-s + (52.7 − 4.59i)12-s + (−48.6 − 39.9i)13-s + (−36.5 − 89.5i)14-s + (−7.78 − 3.22i)15-s + (44.2 + 46.2i)16-s + (64.3 − 26.6i)17-s + ⋯
L(s)  = 1  + (−0.200 + 0.979i)2-s + (−1.12 + 0.600i)3-s + (−0.919 − 0.393i)4-s + (0.0722 + 0.0879i)5-s + (−0.363 − 1.22i)6-s + (−1.53 + 1.02i)7-s + (0.569 − 0.821i)8-s + (0.346 − 0.519i)9-s + (−0.100 + 0.0530i)10-s + (1.24 − 0.377i)11-s + (1.26 − 0.110i)12-s + (−1.03 − 0.851i)13-s + (−0.697 − 1.70i)14-s + (−0.134 − 0.0555i)15-s + (0.690 + 0.722i)16-s + (0.918 − 0.380i)17-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((0.836+0.547i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+3/2) L(s)=((0.836+0.547i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 0.836+0.547i0.836 + 0.547i
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), 0.836+0.547i)(2,\ 128,\ (\ :3/2),\ 0.836 + 0.547i)

Particular Values

L(2)L(2) ≈\approx 0.183061−0.0545711i0.183061 - 0.0545711i
L(12)L(\frac12) ≈\approx 0.183061−0.0545711i0.183061 - 0.0545711i
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+(0.567−2.77i)T 1 + (0.567 - 2.77i)T
good3 1+(5.84−3.12i)T+(15.0−22.4i)T2 1 + (5.84 - 3.12i)T + (15.0 - 22.4i)T^{2}
5 1+(−0.807−0.983i)T+(−24.3+122.i)T2 1 + (-0.807 - 0.983i)T + (-24.3 + 122. i)T^{2}
7 1+(28.4−18.9i)T+(131.−316.i)T2 1 + (28.4 - 18.9i)T + (131. - 316. i)T^{2}
11 1+(−45.4+13.7i)T+(1.10e3−739.i)T2 1 + (-45.4 + 13.7i)T + (1.10e3 - 739. i)T^{2}
13 1+(48.6+39.9i)T+(428.+2.15e3i)T2 1 + (48.6 + 39.9i)T + (428. + 2.15e3i)T^{2}
17 1+(−64.3+26.6i)T+(3.47e3−3.47e3i)T2 1 + (-64.3 + 26.6i)T + (3.47e3 - 3.47e3i)T^{2}
19 1+(−13.3−135.i)T+(−6.72e3+1.33e3i)T2 1 + (-13.3 - 135. i)T + (-6.72e3 + 1.33e3i)T^{2}
23 1+(−19.8+3.95i)T+(1.12e4−4.65e3i)T2 1 + (-19.8 + 3.95i)T + (1.12e4 - 4.65e3i)T^{2}
29 1+(24.6−81.2i)T+(−2.02e4−1.35e4i)T2 1 + (24.6 - 81.2i)T + (-2.02e4 - 1.35e4i)T^{2}
31 1+(73.2+73.2i)T+2.97e4iT2 1 + (73.2 + 73.2i)T + 2.97e4iT^{2}
37 1+(127.+12.5i)T+(4.96e4+9.88e3i)T2 1 + (127. + 12.5i)T + (4.96e4 + 9.88e3i)T^{2}
41 1+(73.2+368.i)T+(−6.36e4+2.63e4i)T2 1 + (73.2 + 368. i)T + (-6.36e4 + 2.63e4i)T^{2}
43 1+(386.+206.i)T+(4.41e4+6.61e4i)T2 1 + (386. + 206. i)T + (4.41e4 + 6.61e4i)T^{2}
47 1+(140.+339.i)T+(−7.34e4+7.34e4i)T2 1 + (140. + 339. i)T + (-7.34e4 + 7.34e4i)T^{2}
53 1+(−128.−422.i)T+(−1.23e5+8.27e4i)T2 1 + (-128. - 422. i)T + (-1.23e5 + 8.27e4i)T^{2}
59 1+(16.9−13.8i)T+(4.00e4−2.01e5i)T2 1 + (16.9 - 13.8i)T + (4.00e4 - 2.01e5i)T^{2}
61 1+(136.+254.i)T+(−1.26e5+1.88e5i)T2 1 + (136. + 254. i)T + (-1.26e5 + 1.88e5i)T^{2}
67 1+(−349.−653.i)T+(−1.67e5+2.50e5i)T2 1 + (-349. - 653. i)T + (-1.67e5 + 2.50e5i)T^{2}
71 1+(−32.1−48.1i)T+(−1.36e5+3.30e5i)T2 1 + (-32.1 - 48.1i)T + (-1.36e5 + 3.30e5i)T^{2}
73 1+(−148.−99.4i)T+(1.48e5+3.59e5i)T2 1 + (-148. - 99.4i)T + (1.48e5 + 3.59e5i)T^{2}
79 1+(−66.9+161.i)T+(−3.48e5−3.48e5i)T2 1 + (-66.9 + 161. i)T + (-3.48e5 - 3.48e5i)T^{2}
83 1+(1.25e3−123.i)T+(5.60e5−1.11e5i)T2 1 + (1.25e3 - 123. i)T + (5.60e5 - 1.11e5i)T^{2}
89 1+(−350.−69.6i)T+(6.51e5+2.69e5i)T2 1 + (-350. - 69.6i)T + (6.51e5 + 2.69e5i)T^{2}
97 1+(533.+533.i)T+9.12e5iT2 1 + (533. + 533. 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

−12.50332782693348291740658855950, −12.02875770340753940001794913095, −10.19854166498352157837674640689, −9.808169510538486976792535029446, −8.604929686209321517138791510200, −6.95687163291693886673944892434, −5.90630843762402536086549330395, −5.38965324612109579143774519075, −3.60273599454953878518291125856, −0.14086606704817761026365495829, 1.20608771388166274230706738817, 3.38635050004815061722647802273, 4.81365165497114795843595671544, 6.53625467421804256672926993397, 7.22009803593463518535105325616, 9.312922814919499038936106398627, 9.842219820041002527710786171089, 11.13647366863753964387009425648, 11.90839111586209618415923627155, 12.74205838500324976595662626040

Graph of the ZZ-function along the critical line