Properties

Label 2-22-11.6-c12-0-9
Degree 22
Conductor 2222
Sign 0.807+0.590i-0.807 + 0.590i
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 + (296. + 215. i)3-s + (1.65e3 − 1.20e3i)4-s + (8.35e3 − 2.57e4i)5-s + (−1.57e4 − 5.13e3i)6-s + (−2.52e4 − 3.47e4i)7-s + (−5.44e4 + 7.49e4i)8-s + (−1.22e5 − 3.77e5i)9-s + 1.22e6i·10-s + (−5.79e5 + 1.67e6i)11-s + 7.51e5·12-s + (5.13e5 − 1.66e5i)13-s + (1.57e6 + 1.14e6i)14-s + (8.02e6 − 5.83e6i)15-s + (1.29e6 − 3.98e6i)16-s + (−1.06e7 − 3.46e6i)17-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (0.407 + 0.295i)3-s + (0.404 − 0.293i)4-s + (0.534 − 1.64i)5-s + (−0.338 − 0.109i)6-s + (−0.214 − 0.295i)7-s + (−0.207 + 0.286i)8-s + (−0.230 − 0.710i)9-s + 1.22i·10-s + (−0.327 + 0.944i)11-s + 0.251·12-s + (0.106 − 0.0345i)13-s + (0.208 + 0.151i)14-s + (0.704 − 0.512i)15-s + (0.0772 − 0.237i)16-s + (−0.442 − 0.143i)17-s + ⋯

Functional equation

Λ(s)=(22s/2ΓC(s)L(s)=((0.807+0.590i)Λ(13s)\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(13-s) \end{aligned}
Λ(s)=(22s/2ΓC(s+6)L(s)=((0.807+0.590i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2222    =    2112 \cdot 11
Sign: 0.807+0.590i-0.807 + 0.590i
Analytic conductor: 20.107820.1078
Root analytic conductor: 4.484174.48417
Motivic weight: 1212
Rational: no
Arithmetic: yes
Character: χ22(17,)\chi_{22} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 22, ( :6), 0.807+0.590i)(2,\ 22,\ (\ :6),\ -0.807 + 0.590i)

Particular Values

L(132)L(\frac{13}{2}) \approx 0.2510340.768998i0.251034 - 0.768998i
L(12)L(\frac12) \approx 0.2510340.768998i0.251034 - 0.768998i
L(7)L(7) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(43.013.9i)T 1 + (43.0 - 13.9i)T
11 1+(5.79e51.67e6i)T 1 + (5.79e5 - 1.67e6i)T
good3 1+(296.215.i)T+(1.64e5+5.05e5i)T2 1 + (-296. - 215. i)T + (1.64e5 + 5.05e5i)T^{2}
5 1+(8.35e3+2.57e4i)T+(1.97e81.43e8i)T2 1 + (-8.35e3 + 2.57e4i)T + (-1.97e8 - 1.43e8i)T^{2}
7 1+(2.52e4+3.47e4i)T+(4.27e9+1.31e10i)T2 1 + (2.52e4 + 3.47e4i)T + (-4.27e9 + 1.31e10i)T^{2}
13 1+(5.13e5+1.66e5i)T+(1.88e131.36e13i)T2 1 + (-5.13e5 + 1.66e5i)T + (1.88e13 - 1.36e13i)T^{2}
17 1+(1.06e7+3.46e6i)T+(4.71e14+3.42e14i)T2 1 + (1.06e7 + 3.46e6i)T + (4.71e14 + 3.42e14i)T^{2}
19 1+(3.35e74.61e7i)T+(6.83e142.10e15i)T2 1 + (3.35e7 - 4.61e7i)T + (-6.83e14 - 2.10e15i)T^{2}
23 1+1.15e8T+2.19e16T2 1 + 1.15e8T + 2.19e16T^{2}
29 1+(5.32e8+7.33e8i)T+(1.09e17+3.36e17i)T2 1 + (5.32e8 + 7.33e8i)T + (-1.09e17 + 3.36e17i)T^{2}
31 1+(4.73e71.45e8i)T+(6.37e17+4.62e17i)T2 1 + (-4.73e7 - 1.45e8i)T + (-6.37e17 + 4.62e17i)T^{2}
37 1+(2.13e9+1.55e9i)T+(2.03e186.26e18i)T2 1 + (-2.13e9 + 1.55e9i)T + (2.03e18 - 6.26e18i)T^{2}
41 1+(1.96e92.70e9i)T+(6.97e182.14e19i)T2 1 + (1.96e9 - 2.70e9i)T + (-6.97e18 - 2.14e19i)T^{2}
43 13.28e9iT3.99e19T2 1 - 3.28e9iT - 3.99e19T^{2}
47 1+(5.64e94.10e9i)T+(3.59e19+1.10e20i)T2 1 + (-5.64e9 - 4.10e9i)T + (3.59e19 + 1.10e20i)T^{2}
53 1+(9.78e9+3.01e10i)T+(3.97e20+2.88e20i)T2 1 + (9.78e9 + 3.01e10i)T + (-3.97e20 + 2.88e20i)T^{2}
59 1+(4.80e103.48e10i)T+(5.49e201.69e21i)T2 1 + (4.80e10 - 3.48e10i)T + (5.49e20 - 1.69e21i)T^{2}
61 1+(3.49e101.13e10i)T+(2.14e21+1.56e21i)T2 1 + (-3.49e10 - 1.13e10i)T + (2.14e21 + 1.56e21i)T^{2}
67 1+1.46e11T+8.18e21T2 1 + 1.46e11T + 8.18e21T^{2}
71 1+(6.00e10+1.84e11i)T+(1.32e229.64e21i)T2 1 + (-6.00e10 + 1.84e11i)T + (-1.32e22 - 9.64e21i)T^{2}
73 1+(3.40e10+4.68e10i)T+(7.07e21+2.17e22i)T2 1 + (3.40e10 + 4.68e10i)T + (-7.07e21 + 2.17e22i)T^{2}
79 1+(3.22e111.04e11i)T+(4.78e223.47e22i)T2 1 + (3.22e11 - 1.04e11i)T + (4.78e22 - 3.47e22i)T^{2}
83 1+(7.18e102.33e10i)T+(8.64e22+6.28e22i)T2 1 + (-7.18e10 - 2.33e10i)T + (8.64e22 + 6.28e22i)T^{2}
89 16.46e11T+2.46e23T2 1 - 6.46e11T + 2.46e23T^{2}
97 1+(4.31e111.32e12i)T+(5.61e23+4.07e23i)T2 1 + (-4.31e11 - 1.32e12i)T + (-5.61e23 + 4.07e23i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−14.87506973336209669202074081819, −13.26134642495403325721127789226, −12.10780140480824980530096140863, −9.992660733334584421761013891557, −9.190747322211950402038886721442, −8.029312550821049401321671525133, −6.00657976831686418842905286797, −4.31271468356864544670733489299, −1.87077108139337935114144751838, −0.31937944388891995245608200814, 2.13694399037144127741965255697, 3.06120496775943806148073082170, 6.09611777322645046776199906198, 7.39010644354115328762151952677, 8.847986879816250884839703464694, 10.47952026178811369184814301912, 11.19547799911218986724384242749, 13.25911329765158068956591467034, 14.30590891234280563920573617123, 15.61361461026954024693995821503

Graph of the ZZ-function along the critical line