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Properties

Label 2-18-9.2-c6-0-3
Degree 22
Conductor 1818
Sign 0.724−0.689i0.724 - 0.689i
Analytic cond. 4.140974.14097
Root an. cond. 2.034932.03493
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.89 + 2.82i)2-s + (25.6 + 8.43i)3-s + (15.9 + 27.7i)4-s + (1.59 − 0.922i)5-s + (101. + 113. i)6-s + (6.34 − 10.9i)7-s + 181. i·8-s + (586. + 432. i)9-s + 10.4·10-s + (−49.8 − 28.7i)11-s + (176. + 845. i)12-s + (−1.41e3 − 2.44e3i)13-s + (62.1 − 35.8i)14-s + (48.7 − 10.1i)15-s + (−512. + 886. i)16-s − 7.22e3i·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.949 + 0.312i)3-s + (0.249 + 0.433i)4-s + (0.0127 − 0.00737i)5-s + (0.471 + 0.527i)6-s + (0.0184 − 0.0320i)7-s + 0.353i·8-s + (0.804 + 0.593i)9-s + 0.0104·10-s + (−0.0374 − 0.0216i)11-s + (0.102 + 0.489i)12-s + (−0.642 − 1.11i)13-s + (0.0226 − 0.0130i)14-s + (0.0144 − 0.00301i)15-s + (−0.125 + 0.216i)16-s − 1.46i·17-s + ⋯

Functional equation

Λ(s)=(18s/2 ΓC(s) L(s)=((0.724−0.689i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(18s/2 ΓC(s+3) L(s)=((0.724−0.689i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1818    =    2⋅322 \cdot 3^{2}
Sign: 0.724−0.689i0.724 - 0.689i
Analytic conductor: 4.140974.14097
Root analytic conductor: 2.034932.03493
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ18(11,⋅)\chi_{18} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 18, ( :3), 0.724−0.689i)(2,\ 18,\ (\ :3),\ 0.724 - 0.689i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 2.36098+0.944083i2.36098 + 0.944083i
L(12)L(\frac12) ≈\approx 2.36098+0.944083i2.36098 + 0.944083i
L(4)L(4) 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+(−4.89−2.82i)T 1 + (-4.89 - 2.82i)T
3 1+(−25.6−8.43i)T 1 + (-25.6 - 8.43i)T
good5 1+(−1.59+0.922i)T+(7.81e3−1.35e4i)T2 1 + (-1.59 + 0.922i)T + (7.81e3 - 1.35e4i)T^{2}
7 1+(−6.34+10.9i)T+(−5.88e4−1.01e5i)T2 1 + (-6.34 + 10.9i)T + (-5.88e4 - 1.01e5i)T^{2}
11 1+(49.8+28.7i)T+(8.85e5+1.53e6i)T2 1 + (49.8 + 28.7i)T + (8.85e5 + 1.53e6i)T^{2}
13 1+(1.41e3+2.44e3i)T+(−2.41e6+4.18e6i)T2 1 + (1.41e3 + 2.44e3i)T + (-2.41e6 + 4.18e6i)T^{2}
17 1+7.22e3iT−2.41e7T2 1 + 7.22e3iT - 2.41e7T^{2}
19 1+1.06e4T+4.70e7T2 1 + 1.06e4T + 4.70e7T^{2}
23 1+(−1.15e4+6.68e3i)T+(7.40e7−1.28e8i)T2 1 + (-1.15e4 + 6.68e3i)T + (7.40e7 - 1.28e8i)T^{2}
29 1+(−3.07e4−1.77e4i)T+(2.97e8+5.15e8i)T2 1 + (-3.07e4 - 1.77e4i)T + (2.97e8 + 5.15e8i)T^{2}
31 1+(−8.65e3−1.49e4i)T+(−4.43e8+7.68e8i)T2 1 + (-8.65e3 - 1.49e4i)T + (-4.43e8 + 7.68e8i)T^{2}
37 1+6.04e4T+2.56e9T2 1 + 6.04e4T + 2.56e9T^{2}
41 1+(7.54e4−4.35e4i)T+(2.37e9−4.11e9i)T2 1 + (7.54e4 - 4.35e4i)T + (2.37e9 - 4.11e9i)T^{2}
43 1+(−3.79e4+6.57e4i)T+(−3.16e9−5.47e9i)T2 1 + (-3.79e4 + 6.57e4i)T + (-3.16e9 - 5.47e9i)T^{2}
47 1+(−6.18e4−3.57e4i)T+(5.38e9+9.33e9i)T2 1 + (-6.18e4 - 3.57e4i)T + (5.38e9 + 9.33e9i)T^{2}
53 1−1.47e5iT−2.21e10T2 1 - 1.47e5iT - 2.21e10T^{2}
59 1+(−1.00e5+5.78e4i)T+(2.10e10−3.65e10i)T2 1 + (-1.00e5 + 5.78e4i)T + (2.10e10 - 3.65e10i)T^{2}
61 1+(−2.01e4+3.48e4i)T+(−2.57e10−4.46e10i)T2 1 + (-2.01e4 + 3.48e4i)T + (-2.57e10 - 4.46e10i)T^{2}
67 1+(−9.08e4−1.57e5i)T+(−4.52e10+7.83e10i)T2 1 + (-9.08e4 - 1.57e5i)T + (-4.52e10 + 7.83e10i)T^{2}
71 1+3.01e5iT−1.28e11T2 1 + 3.01e5iT - 1.28e11T^{2}
73 1+6.12e5T+1.51e11T2 1 + 6.12e5T + 1.51e11T^{2}
79 1+(−1.71e5+2.97e5i)T+(−1.21e11−2.10e11i)T2 1 + (-1.71e5 + 2.97e5i)T + (-1.21e11 - 2.10e11i)T^{2}
83 1+(1.31e5+7.59e4i)T+(1.63e11+2.83e11i)T2 1 + (1.31e5 + 7.59e4i)T + (1.63e11 + 2.83e11i)T^{2}
89 1+1.02e5iT−4.96e11T2 1 + 1.02e5iT - 4.96e11T^{2}
97 1+(−2.47e5+4.29e5i)T+(−4.16e11−7.21e11i)T2 1 + (-2.47e5 + 4.29e5i)T + (-4.16e11 - 7.21e11i)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

−17.29392126391587615726051532915, −15.80532557520805129867838457093, −14.88453984207508013941199817084, −13.72753207912969945392100949253, −12.51116138468688426295457208307, −10.47225791968756840191232791848, −8.729480083695416397887817718889, −7.21167299158374884369317297291, −4.86317093346278517272727683692, −2.89893191550281573396473599301, 2.10492963411147185629111956060, 4.13649598353590919107784415860, 6.65161925255404150004090982820, 8.547007156034662046228693818677, 10.20234104803373087620740184412, 12.07530547442694177273090832266, 13.25550783972698840405812192850, 14.43893579050471927713847743808, 15.39803937252827189608268853609, 17.24658019521784782259658905517

Graph of the ZZ-function along the critical line