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Properties

Label 2-2e7-128.101-c3-0-33
Degree 22
Conductor 128128
Sign 0.168+0.985i0.168 + 0.985i
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  + (−2.42 − 1.46i)2-s + (7.50 − 4.01i)3-s + (3.71 + 7.08i)4-s + (0.941 + 1.14i)5-s + (−24.0 − 1.27i)6-s + (2.81 − 1.88i)7-s + (1.37 − 22.5i)8-s + (25.2 − 37.7i)9-s + (−0.599 − 4.15i)10-s + (28.5 − 8.66i)11-s + (56.3 + 38.2i)12-s + (−2.68 − 2.20i)13-s + (−9.56 + 0.432i)14-s + (11.6 + 4.83i)15-s + (−36.3 + 52.6i)16-s + (62.9 − 26.0i)17-s + ⋯
L(s)  = 1  + (−0.855 − 0.517i)2-s + (1.44 − 0.771i)3-s + (0.464 + 0.885i)4-s + (0.0842 + 0.102i)5-s + (−1.63 − 0.0867i)6-s + (0.151 − 0.101i)7-s + (0.0607 − 0.998i)8-s + (0.934 − 1.39i)9-s + (−0.0189 − 0.131i)10-s + (0.782 − 0.237i)11-s + (1.35 + 0.920i)12-s + (−0.0573 − 0.0470i)13-s + (−0.182 + 0.00825i)14-s + (0.200 + 0.0831i)15-s + (−0.568 + 0.822i)16-s + (0.898 − 0.372i)17-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((0.168+0.985i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+3/2) L(s)=((0.168+0.985i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 0.168+0.985i0.168 + 0.985i
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), 0.168+0.985i)(2,\ 128,\ (\ :3/2),\ 0.168 + 0.985i)

Particular Values

L(2)L(2) ≈\approx 1.41621−1.19502i1.41621 - 1.19502i
L(12)L(\frac12) ≈\approx 1.41621−1.19502i1.41621 - 1.19502i
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+(2.42+1.46i)T 1 + (2.42 + 1.46i)T
good3 1+(−7.50+4.01i)T+(15.0−22.4i)T2 1 + (-7.50 + 4.01i)T + (15.0 - 22.4i)T^{2}
5 1+(−0.941−1.14i)T+(−24.3+122.i)T2 1 + (-0.941 - 1.14i)T + (-24.3 + 122. i)T^{2}
7 1+(−2.81+1.88i)T+(131.−316.i)T2 1 + (-2.81 + 1.88i)T + (131. - 316. i)T^{2}
11 1+(−28.5+8.66i)T+(1.10e3−739.i)T2 1 + (-28.5 + 8.66i)T + (1.10e3 - 739. i)T^{2}
13 1+(2.68+2.20i)T+(428.+2.15e3i)T2 1 + (2.68 + 2.20i)T + (428. + 2.15e3i)T^{2}
17 1+(−62.9+26.0i)T+(3.47e3−3.47e3i)T2 1 + (-62.9 + 26.0i)T + (3.47e3 - 3.47e3i)T^{2}
19 1+(−1.70−17.3i)T+(−6.72e3+1.33e3i)T2 1 + (-1.70 - 17.3i)T + (-6.72e3 + 1.33e3i)T^{2}
23 1+(83.5−16.6i)T+(1.12e4−4.65e3i)T2 1 + (83.5 - 16.6i)T + (1.12e4 - 4.65e3i)T^{2}
29 1+(−38.0+125.i)T+(−2.02e4−1.35e4i)T2 1 + (-38.0 + 125. i)T + (-2.02e4 - 1.35e4i)T^{2}
31 1+(−29.8−29.8i)T+2.97e4iT2 1 + (-29.8 - 29.8i)T + 2.97e4iT^{2}
37 1+(301.+29.6i)T+(4.96e4+9.88e3i)T2 1 + (301. + 29.6i)T + (4.96e4 + 9.88e3i)T^{2}
41 1+(−45.5−228.i)T+(−6.36e4+2.63e4i)T2 1 + (-45.5 - 228. i)T + (-6.36e4 + 2.63e4i)T^{2}
43 1+(328.+175.i)T+(4.41e4+6.61e4i)T2 1 + (328. + 175. i)T + (4.41e4 + 6.61e4i)T^{2}
47 1+(−114.−277.i)T+(−7.34e4+7.34e4i)T2 1 + (-114. - 277. i)T + (-7.34e4 + 7.34e4i)T^{2}
53 1+(−147.−484.i)T+(−1.23e5+8.27e4i)T2 1 + (-147. - 484. i)T + (-1.23e5 + 8.27e4i)T^{2}
59 1+(−219.+180.i)T+(4.00e4−2.01e5i)T2 1 + (-219. + 180. i)T + (4.00e4 - 2.01e5i)T^{2}
61 1+(−261.−488.i)T+(−1.26e5+1.88e5i)T2 1 + (-261. - 488. i)T + (-1.26e5 + 1.88e5i)T^{2}
67 1+(166.+312.i)T+(−1.67e5+2.50e5i)T2 1 + (166. + 312. i)T + (-1.67e5 + 2.50e5i)T^{2}
71 1+(158.+237.i)T+(−1.36e5+3.30e5i)T2 1 + (158. + 237. i)T + (-1.36e5 + 3.30e5i)T^{2}
73 1+(−767.−513.i)T+(1.48e5+3.59e5i)T2 1 + (-767. - 513. i)T + (1.48e5 + 3.59e5i)T^{2}
79 1+(438.−1.05e3i)T+(−3.48e5−3.48e5i)T2 1 + (438. - 1.05e3i)T + (-3.48e5 - 3.48e5i)T^{2}
83 1+(−833.+82.0i)T+(5.60e5−1.11e5i)T2 1 + (-833. + 82.0i)T + (5.60e5 - 1.11e5i)T^{2}
89 1+(−819.−163.i)T+(6.51e5+2.69e5i)T2 1 + (-819. - 163. i)T + (6.51e5 + 2.69e5i)T^{2}
97 1+(1.29e3+1.29e3i)T+9.12e5iT2 1 + (1.29e3 + 1.29e3i)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

−12.50874975015314414997281477128, −11.77787858036060194724662074601, −10.25391616016619138858669402028, −9.325102550816599188418121646220, −8.369536631933761425734694407422, −7.65188248814715372783493878206, −6.53527937318868760678953992738, −3.79522361822244153077840366705, −2.60606401842715177502735148109, −1.26617864794557674374754198849, 1.84043819493280665407115236743, 3.55327102369216168044272060062, 5.17579060069284398422414386283, 6.89590820389966844574737907766, 8.107689277872858440915517348496, 8.842115996376247822631050631563, 9.690986839258311344564445133481, 10.45921467937348297369232424235, 11.90422216001183887157486385100, 13.55769592909778495108476766335

Graph of the ZZ-function along the critical line