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Properties

Label 2-22-11.8-c12-0-10
Degree 22
Conductor 2222
Sign −0.971+0.237i-0.971 + 0.237i
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  + (26.6 + 36.6i)2-s + (−215. − 662. i)3-s + (−632. + 1.94e3i)4-s + (−6.73e3 − 4.88e3i)5-s + (1.85e4 − 2.54e4i)6-s + (1.27e5 + 4.14e4i)7-s + (−8.81e4 + 2.86e4i)8-s + (3.75e4 − 2.72e4i)9-s − 3.76e5i·10-s + (−1.74e6 + 2.99e5i)11-s + 1.42e6·12-s + (−3.66e6 − 5.03e6i)13-s + (1.87e6 + 5.76e6i)14-s + (−1.79e6 + 5.51e6i)15-s + (−3.39e6 − 2.46e6i)16-s + (−2.22e7 + 3.06e7i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.295 − 0.908i)3-s + (−0.154 + 0.475i)4-s + (−0.430 − 0.312i)5-s + (0.397 − 0.546i)6-s + (1.08 + 0.351i)7-s + (−0.336 + 0.109i)8-s + (0.0706 − 0.0513i)9-s − 0.376i·10-s + (−0.985 + 0.169i)11-s + 0.477·12-s + (−0.758 − 1.04i)13-s + (0.248 + 0.765i)14-s + (−0.157 + 0.483i)15-s + (−0.202 − 0.146i)16-s + (−0.921 + 1.26i)17-s + ⋯

Functional equation

Λ(s)=(22s/2 ΓC(s) L(s)=((−0.971+0.237i) Λ‾(13−s)\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(13-s) \end{aligned}
Λ(s)=(22s/2 ΓC(s+6) L(s)=((−0.971+0.237i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2222    =    2⋅112 \cdot 11
Sign: −0.971+0.237i-0.971 + 0.237i
Analytic conductor: 20.107820.1078
Root analytic conductor: 4.484174.48417
Motivic weight: 1212
Rational: no
Arithmetic: yes
Character: χ22(19,⋅)\chi_{22} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 22, ( :6), −0.971+0.237i)(2,\ 22,\ (\ :6),\ -0.971 + 0.237i)

Particular Values

L(132)L(\frac{13}{2}) ≈\approx 0.0299987−0.249243i0.0299987 - 0.249243i
L(12)L(\frac12) ≈\approx 0.0299987−0.249243i0.0299987 - 0.249243i
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+(−26.6−36.6i)T 1 + (-26.6 - 36.6i)T
11 1+(1.74e6−2.99e5i)T 1 + (1.74e6 - 2.99e5i)T
good3 1+(215.+662.i)T+(−4.29e5+3.12e5i)T2 1 + (215. + 662. i)T + (-4.29e5 + 3.12e5i)T^{2}
5 1+(6.73e3+4.88e3i)T+(7.54e7+2.32e8i)T2 1 + (6.73e3 + 4.88e3i)T + (7.54e7 + 2.32e8i)T^{2}
7 1+(−1.27e5−4.14e4i)T+(1.11e10+8.13e9i)T2 1 + (-1.27e5 - 4.14e4i)T + (1.11e10 + 8.13e9i)T^{2}
13 1+(3.66e6+5.03e6i)T+(−7.19e12+2.21e13i)T2 1 + (3.66e6 + 5.03e6i)T + (-7.19e12 + 2.21e13i)T^{2}
17 1+(2.22e7−3.06e7i)T+(−1.80e14−5.54e14i)T2 1 + (2.22e7 - 3.06e7i)T + (-1.80e14 - 5.54e14i)T^{2}
19 1+(6.98e7−2.26e7i)T+(1.79e15−1.30e15i)T2 1 + (6.98e7 - 2.26e7i)T + (1.79e15 - 1.30e15i)T^{2}
23 1−7.00e7T+2.19e16T2 1 - 7.00e7T + 2.19e16T^{2}
29 1+(4.18e8+1.36e8i)T+(2.86e17+2.07e17i)T2 1 + (4.18e8 + 1.36e8i)T + (2.86e17 + 2.07e17i)T^{2}
31 1+(6.96e8−5.06e8i)T+(2.43e17−7.49e17i)T2 1 + (6.96e8 - 5.06e8i)T + (2.43e17 - 7.49e17i)T^{2}
37 1+(3.07e8−9.46e8i)T+(−5.32e18−3.86e18i)T2 1 + (3.07e8 - 9.46e8i)T + (-5.32e18 - 3.86e18i)T^{2}
41 1+(−8.21e9+2.66e9i)T+(1.82e19−1.32e19i)T2 1 + (-8.21e9 + 2.66e9i)T + (1.82e19 - 1.32e19i)T^{2}
43 1+2.31e9iT−3.99e19T2 1 + 2.31e9iT - 3.99e19T^{2}
47 1+(−3.00e9−9.25e9i)T+(−9.40e19+6.82e19i)T2 1 + (-3.00e9 - 9.25e9i)T + (-9.40e19 + 6.82e19i)T^{2}
53 1+(1.62e10−1.17e10i)T+(1.51e20−4.67e20i)T2 1 + (1.62e10 - 1.17e10i)T + (1.51e20 - 4.67e20i)T^{2}
59 1+(−1.78e10+5.49e10i)T+(−1.43e21−1.04e21i)T2 1 + (-1.78e10 + 5.49e10i)T + (-1.43e21 - 1.04e21i)T^{2}
61 1+(2.93e10−4.04e10i)T+(−8.20e20−2.52e21i)T2 1 + (2.93e10 - 4.04e10i)T + (-8.20e20 - 2.52e21i)T^{2}
67 1+5.55e10T+8.18e21T2 1 + 5.55e10T + 8.18e21T^{2}
71 1+(−6.21e10−4.51e10i)T+(5.07e21+1.56e22i)T2 1 + (-6.21e10 - 4.51e10i)T + (5.07e21 + 1.56e22i)T^{2}
73 1+(−3.46e10−1.12e10i)T+(1.85e22+1.34e22i)T2 1 + (-3.46e10 - 1.12e10i)T + (1.85e22 + 1.34e22i)T^{2}
79 1+(1.91e11+2.64e11i)T+(−1.82e22+5.61e22i)T2 1 + (1.91e11 + 2.64e11i)T + (-1.82e22 + 5.61e22i)T^{2}
83 1+(−2.22e11+3.06e11i)T+(−3.30e22−1.01e23i)T2 1 + (-2.22e11 + 3.06e11i)T + (-3.30e22 - 1.01e23i)T^{2}
89 1+3.32e11T+2.46e23T2 1 + 3.32e11T + 2.46e23T^{2}
97 1+(7.24e10−5.26e10i)T+(2.14e23−6.59e23i)T2 1 + (7.24e10 - 5.26e10i)T + (2.14e23 - 6.59e23i)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

−14.73545126755452283553350417096, −12.91330956298723200681030539424, −12.45919093223207520641765086189, −10.81659853054700554263124122021, −8.373567103378181406416518358206, −7.53037341247914220783407400975, −5.89798991153869137380568061471, −4.46610136752527754743884550922, −2.04292902647804332503375069262, −0.07626840740642007664936533243, 2.22841389889385031054889336502, 4.22934567734554247751284294742, 5.01743199784209421607402463268, 7.35450807502403691337769147739, 9.317775226578844531139052473091, 10.90247720948665435994881864732, 11.27838525985393666641114105063, 13.13581696834715465573460493156, 14.55971766954604131303926006942, 15.50481964756676447582829724144

Graph of the ZZ-function along the critical line