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Properties

Label 2-18-9.5-c10-0-0
Degree 22
Conductor 1818
Sign −0.948−0.316i-0.948 - 0.316i
Analytic cond. 11.436411.4364
Root an. cond. 3.381773.38177
Motivic weight 1010
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−19.5 + 11.3i)2-s + (208. − 124. i)3-s + (255. − 443. i)4-s + (−4.47e3 − 2.58e3i)5-s + (−2.68e3 + 4.79e3i)6-s + (7.93e3 + 1.37e4i)7-s + 1.15e4i·8-s + (2.81e4 − 5.19e4i)9-s + 1.16e5·10-s + (−1.21e5 + 6.99e4i)11-s + (−1.70e3 − 1.24e5i)12-s + (−3.43e5 + 5.95e5i)13-s + (−3.11e5 − 1.79e5i)14-s + (−1.25e6 + 1.71e4i)15-s + (−1.31e5 − 2.27e5i)16-s − 9.11e5i·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.859 − 0.511i)3-s + (0.249 − 0.433i)4-s + (−1.43 − 0.826i)5-s + (−0.345 + 0.617i)6-s + (0.472 + 0.818i)7-s + 0.353i·8-s + (0.476 − 0.879i)9-s + 1.16·10-s + (−0.752 + 0.434i)11-s + (−0.00683 − 0.499i)12-s + (−0.925 + 1.60i)13-s + (−0.578 − 0.334i)14-s + (−1.65 + 0.0225i)15-s + (−0.125 − 0.216i)16-s − 0.642i·17-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((−0.948−0.316i) Λ‾(11−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(11-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+5) L(s)=((−0.948−0.316i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: −0.948−0.316i-0.948 - 0.316i
Analytic conductor: 11.436411.4364
Root analytic conductor: 3.381773.38177
Motivic weight: 1010
Rational: no
Arithmetic: yes
Character: χ18(5,⋅)\chi_{18} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 18, ( :5), −0.948−0.316i)(2,\ 18,\ (\ :5),\ -0.948 - 0.316i)

Particular Values

L(112)L(\frac{11}{2}) ≈\approx 0.0162099+0.0999009i0.0162099 + 0.0999009i
L(12)L(\frac12) ≈\approx 0.0162099+0.0999009i0.0162099 + 0.0999009i
L(6)L(6) 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+(19.5−11.3i)T 1 + (19.5 - 11.3i)T
3 1+(−208.+124.i)T 1 + (-208. + 124. i)T
good5 1+(4.47e3+2.58e3i)T+(4.88e6+8.45e6i)T2 1 + (4.47e3 + 2.58e3i)T + (4.88e6 + 8.45e6i)T^{2}
7 1+(−7.93e3−1.37e4i)T+(−1.41e8+2.44e8i)T2 1 + (-7.93e3 - 1.37e4i)T + (-1.41e8 + 2.44e8i)T^{2}
11 1+(1.21e5−6.99e4i)T+(1.29e10−2.24e10i)T2 1 + (1.21e5 - 6.99e4i)T + (1.29e10 - 2.24e10i)T^{2}
13 1+(3.43e5−5.95e5i)T+(−6.89e10−1.19e11i)T2 1 + (3.43e5 - 5.95e5i)T + (-6.89e10 - 1.19e11i)T^{2}
17 1+9.11e5iT−2.01e12T2 1 + 9.11e5iT - 2.01e12T^{2}
19 1+3.96e6T+6.13e12T2 1 + 3.96e6T + 6.13e12T^{2}
23 1+(2.79e5+1.61e5i)T+(2.07e13+3.58e13i)T2 1 + (2.79e5 + 1.61e5i)T + (2.07e13 + 3.58e13i)T^{2}
29 1+(1.96e7−1.13e7i)T+(2.10e14−3.64e14i)T2 1 + (1.96e7 - 1.13e7i)T + (2.10e14 - 3.64e14i)T^{2}
31 1+(4.80e5−8.32e5i)T+(−4.09e14−7.09e14i)T2 1 + (4.80e5 - 8.32e5i)T + (-4.09e14 - 7.09e14i)T^{2}
37 1−8.29e7T+4.80e15T2 1 - 8.29e7T + 4.80e15T^{2}
41 1+(4.29e7+2.47e7i)T+(6.71e15+1.16e16i)T2 1 + (4.29e7 + 2.47e7i)T + (6.71e15 + 1.16e16i)T^{2}
43 1+(1.64e7+2.85e7i)T+(−1.08e16+1.87e16i)T2 1 + (1.64e7 + 2.85e7i)T + (-1.08e16 + 1.87e16i)T^{2}
47 1+(8.48e6−4.89e6i)T+(2.62e16−4.55e16i)T2 1 + (8.48e6 - 4.89e6i)T + (2.62e16 - 4.55e16i)T^{2}
53 1+1.50e8iT−1.74e17T2 1 + 1.50e8iT - 1.74e17T^{2}
59 1+(6.48e8+3.74e8i)T+(2.55e17+4.42e17i)T2 1 + (6.48e8 + 3.74e8i)T + (2.55e17 + 4.42e17i)T^{2}
61 1+(3.97e7+6.88e7i)T+(−3.56e17+6.17e17i)T2 1 + (3.97e7 + 6.88e7i)T + (-3.56e17 + 6.17e17i)T^{2}
67 1+(1.31e8−2.28e8i)T+(−9.11e17−1.57e18i)T2 1 + (1.31e8 - 2.28e8i)T + (-9.11e17 - 1.57e18i)T^{2}
71 1−1.74e8iT−3.25e18T2 1 - 1.74e8iT - 3.25e18T^{2}
73 1−1.37e9T+4.29e18T2 1 - 1.37e9T + 4.29e18T^{2}
79 1+(2.41e9+4.18e9i)T+(−4.73e18+8.19e18i)T2 1 + (2.41e9 + 4.18e9i)T + (-4.73e18 + 8.19e18i)T^{2}
83 1+(4.80e9−2.77e9i)T+(7.75e18−1.34e19i)T2 1 + (4.80e9 - 2.77e9i)T + (7.75e18 - 1.34e19i)T^{2}
89 1−3.34e9iT−3.11e19T2 1 - 3.34e9iT - 3.11e19T^{2}
97 1+(6.16e9+1.06e10i)T+(−3.68e19+6.38e19i)T2 1 + (6.16e9 + 1.06e10i)T + (-3.68e19 + 6.38e19i)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

−16.73786222115780304334020839846, −15.47678417101076356204985783358, −14.64767162795425803994097411050, −12.66246827985630516099475062730, −11.62493674682044567297494143098, −9.221910827232684382047585960141, −8.280438764903187959201137586095, −7.18757938313609848096400104292, −4.54531880389023683113858412803, −2.06088104474018727677836216144, 0.04710630738326260869514210760, 2.79877334379376025231418722940, 4.12086723678550276284513452917, 7.61068481017722376140944089740, 8.140138999694224419837418880051, 10.34037284498689335669540411951, 10.97968718795100662245284974077, 12.91390390203576059796545398342, 14.79873966431005221501318211970, 15.44282588199116987716328553446

Graph of the ZZ-function along the critical line