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Properties

Label 2-22-11.2-c12-0-9
Degree 22
Conductor 2222
Sign 0.585−0.810i0.585 - 0.810i
Analytic cond. 20.107820.1078
Root an. cond. 4.484174.48417
Motivic weight 1212
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (43.0 + 13.9i)2-s + (232. − 168. i)3-s + (1.65e3 + 1.20e3i)4-s + (6.94e3 + 2.13e4i)5-s + (1.23e4 − 4.01e3i)6-s + (1.27e5 − 1.76e5i)7-s + (5.44e4 + 7.49e4i)8-s + (−1.38e5 + 4.27e5i)9-s + 1.01e6i·10-s + (−1.67e6 + 5.89e5i)11-s + 5.87e5·12-s + (3.06e6 + 9.95e5i)13-s + (7.96e6 − 5.78e6i)14-s + (5.22e6 + 3.79e6i)15-s + (1.29e6 + 3.98e6i)16-s + (4.69e6 − 1.52e6i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.318 − 0.231i)3-s + (0.404 + 0.293i)4-s + (0.444 + 1.36i)5-s + (0.264 − 0.0860i)6-s + (1.08 − 1.49i)7-s + (0.207 + 0.286i)8-s + (−0.261 + 0.803i)9-s + 1.01i·10-s + (−0.943 + 0.332i)11-s + 0.196·12-s + (0.634 + 0.206i)13-s + (1.05 − 0.768i)14-s + (0.458 + 0.333i)15-s + (0.0772 + 0.237i)16-s + (0.194 − 0.0632i)17-s + ⋯

Functional equation

Λ(s)=(22s/2 ΓC(s) L(s)=((0.585−0.810i) Λ‾(13−s)\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(13-s) \end{aligned}
Λ(s)=(22s/2 ΓC(s+6) L(s)=((0.585−0.810i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2222    =    2⋅112 \cdot 11
Sign: 0.585−0.810i0.585 - 0.810i
Analytic conductor: 20.107820.1078
Root analytic conductor: 4.484174.48417
Motivic weight: 1212
Rational: no
Arithmetic: yes
Character: χ22(13,⋅)\chi_{22} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 22, ( :6), 0.585−0.810i)(2,\ 22,\ (\ :6),\ 0.585 - 0.810i)

Particular Values

L(132)L(\frac{13}{2}) ≈\approx 3.28301+1.67894i3.28301 + 1.67894i
L(12)L(\frac12) ≈\approx 3.28301+1.67894i3.28301 + 1.67894i
L(7)L(7) 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+(−43.0−13.9i)T 1 + (-43.0 - 13.9i)T
11 1+(1.67e6−5.89e5i)T 1 + (1.67e6 - 5.89e5i)T
good3 1+(−232.+168.i)T+(1.64e5−5.05e5i)T2 1 + (-232. + 168. i)T + (1.64e5 - 5.05e5i)T^{2}
5 1+(−6.94e3−2.13e4i)T+(−1.97e8+1.43e8i)T2 1 + (-6.94e3 - 2.13e4i)T + (-1.97e8 + 1.43e8i)T^{2}
7 1+(−1.27e5+1.76e5i)T+(−4.27e9−1.31e10i)T2 1 + (-1.27e5 + 1.76e5i)T + (-4.27e9 - 1.31e10i)T^{2}
13 1+(−3.06e6−9.95e5i)T+(1.88e13+1.36e13i)T2 1 + (-3.06e6 - 9.95e5i)T + (1.88e13 + 1.36e13i)T^{2}
17 1+(−4.69e6+1.52e6i)T+(4.71e14−3.42e14i)T2 1 + (-4.69e6 + 1.52e6i)T + (4.71e14 - 3.42e14i)T^{2}
19 1+(−3.04e7−4.18e7i)T+(−6.83e14+2.10e15i)T2 1 + (-3.04e7 - 4.18e7i)T + (-6.83e14 + 2.10e15i)T^{2}
23 1−2.19e8T+2.19e16T2 1 - 2.19e8T + 2.19e16T^{2}
29 1+(3.99e8−5.49e8i)T+(−1.09e17−3.36e17i)T2 1 + (3.99e8 - 5.49e8i)T + (-1.09e17 - 3.36e17i)T^{2}
31 1+(−1.45e8+4.47e8i)T+(−6.37e17−4.62e17i)T2 1 + (-1.45e8 + 4.47e8i)T + (-6.37e17 - 4.62e17i)T^{2}
37 1+(2.11e9+1.53e9i)T+(2.03e18+6.26e18i)T2 1 + (2.11e9 + 1.53e9i)T + (2.03e18 + 6.26e18i)T^{2}
41 1+(−1.01e9−1.39e9i)T+(−6.97e18+2.14e19i)T2 1 + (-1.01e9 - 1.39e9i)T + (-6.97e18 + 2.14e19i)T^{2}
43 1+7.35e9iT−3.99e19T2 1 + 7.35e9iT - 3.99e19T^{2}
47 1+(9.14e9−6.64e9i)T+(3.59e19−1.10e20i)T2 1 + (9.14e9 - 6.64e9i)T + (3.59e19 - 1.10e20i)T^{2}
53 1+(−1.11e10+3.43e10i)T+(−3.97e20−2.88e20i)T2 1 + (-1.11e10 + 3.43e10i)T + (-3.97e20 - 2.88e20i)T^{2}
59 1+(2.56e10+1.86e10i)T+(5.49e20+1.69e21i)T2 1 + (2.56e10 + 1.86e10i)T + (5.49e20 + 1.69e21i)T^{2}
61 1+(3.41e10−1.10e10i)T+(2.14e21−1.56e21i)T2 1 + (3.41e10 - 1.10e10i)T + (2.14e21 - 1.56e21i)T^{2}
67 1−9.79e10T+8.18e21T2 1 - 9.79e10T + 8.18e21T^{2}
71 1+(5.90e10+1.81e11i)T+(−1.32e22+9.64e21i)T2 1 + (5.90e10 + 1.81e11i)T + (-1.32e22 + 9.64e21i)T^{2}
73 1+(−4.51e10+6.21e10i)T+(−7.07e21−2.17e22i)T2 1 + (-4.51e10 + 6.21e10i)T + (-7.07e21 - 2.17e22i)T^{2}
79 1+(1.85e11+6.02e10i)T+(4.78e22+3.47e22i)T2 1 + (1.85e11 + 6.02e10i)T + (4.78e22 + 3.47e22i)T^{2}
83 1+(−2.73e11+8.88e10i)T+(8.64e22−6.28e22i)T2 1 + (-2.73e11 + 8.88e10i)T + (8.64e22 - 6.28e22i)T^{2}
89 1−4.38e11T+2.46e23T2 1 - 4.38e11T + 2.46e23T^{2}
97 1+(2.13e11−6.57e11i)T+(−5.61e23−4.07e23i)T2 1 + (2.13e11 - 6.57e11i)T + (-5.61e23 - 4.07e23i)T^{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

−14.80553486565413086965927660183, −14.06620973596091708632509653478, −13.31394793713968394682149117379, −11.07753891282438794126305613752, −10.51743239307205734987611406464, −7.83404600535012754697586392344, −7.04867379046231647846732724417, −5.15201729416173564456342994248, −3.35791956639433298784937440541, −1.80683490473302409358034834060, 1.17483028671454424826175896376, 2.79504475201254759515794715140, 4.90495260705984826535019268183, 5.68463860853567114321838349943, 8.393416693837112255943037043655, 9.246447172136348137690960763496, 11.32960900182945603285526583760, 12.43778286010381969099654366555, 13.51930693204702331069878823058, 15.01864455563466952138888147253

Graph of the ZZ-function along the critical line