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Properties

Label 2-18e2-9.2-c8-0-22
Degree 22
Conductor 324324
Sign −0.642+0.766i-0.642 + 0.766i
Analytic cond. 131.990131.990
Root an. cond. 11.488711.4887
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−980. + 566. i)5-s + (1.54e3 − 2.67e3i)7-s + (980. + 566. i)11-s + (3.64e3 + 6.31e3i)13-s + 5.89e4i·17-s − 8.03e4·19-s + (−8.43e4 + 4.87e4i)23-s + (4.46e5 − 7.72e5i)25-s + (7.48e5 + 4.32e5i)29-s + (−2.17e5 − 3.77e5i)31-s + 3.50e6i·35-s + 1.15e6·37-s + (−2.35e6 + 1.35e6i)41-s + (−4.95e5 + 8.57e5i)43-s + (5.80e6 + 3.35e6i)47-s + ⋯
L(s)  = 1  + (−1.56 + 0.906i)5-s + (0.644 − 1.11i)7-s + (0.0670 + 0.0386i)11-s + (0.127 + 0.221i)13-s + 0.705i·17-s − 0.616·19-s + (−0.301 + 0.174i)23-s + (1.14 − 1.97i)25-s + (1.05 + 0.610i)29-s + (−0.236 − 0.408i)31-s + 2.33i·35-s + 0.618·37-s + (−0.832 + 0.480i)41-s + (−0.144 + 0.250i)43-s + (1.18 + 0.686i)47-s + ⋯

Functional equation

Λ(s)=(324s/2 ΓC(s) L(s)=((−0.642+0.766i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(324s/2 ΓC(s+4) L(s)=((−0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 324324    =    22⋅342^{2} \cdot 3^{4}
Sign: −0.642+0.766i-0.642 + 0.766i
Analytic conductor: 131.990131.990
Root analytic conductor: 11.488711.4887
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ324(269,⋅)\chi_{324} (269, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 324, ( :4), −0.642+0.766i)(2,\ 324,\ (\ :4),\ -0.642 + 0.766i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.31821811390.3182181139
L(12)L(\frac12) ≈\approx 0.31821811390.3182181139
L(5)L(5) 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 1
3 1 1
good5 1+(980.−566.i)T+(1.95e5−3.38e5i)T2 1 + (980. - 566. i)T + (1.95e5 - 3.38e5i)T^{2}
7 1+(−1.54e3+2.67e3i)T+(−2.88e6−4.99e6i)T2 1 + (-1.54e3 + 2.67e3i)T + (-2.88e6 - 4.99e6i)T^{2}
11 1+(−980.−566.i)T+(1.07e8+1.85e8i)T2 1 + (-980. - 566. i)T + (1.07e8 + 1.85e8i)T^{2}
13 1+(−3.64e3−6.31e3i)T+(−4.07e8+7.06e8i)T2 1 + (-3.64e3 - 6.31e3i)T + (-4.07e8 + 7.06e8i)T^{2}
17 1−5.89e4iT−6.97e9T2 1 - 5.89e4iT - 6.97e9T^{2}
19 1+8.03e4T+1.69e10T2 1 + 8.03e4T + 1.69e10T^{2}
23 1+(8.43e4−4.87e4i)T+(3.91e10−6.78e10i)T2 1 + (8.43e4 - 4.87e4i)T + (3.91e10 - 6.78e10i)T^{2}
29 1+(−7.48e5−4.32e5i)T+(2.50e11+4.33e11i)T2 1 + (-7.48e5 - 4.32e5i)T + (2.50e11 + 4.33e11i)T^{2}
31 1+(2.17e5+3.77e5i)T+(−4.26e11+7.38e11i)T2 1 + (2.17e5 + 3.77e5i)T + (-4.26e11 + 7.38e11i)T^{2}
37 1−1.15e6T+3.51e12T2 1 - 1.15e6T + 3.51e12T^{2}
41 1+(2.35e6−1.35e6i)T+(3.99e12−6.91e12i)T2 1 + (2.35e6 - 1.35e6i)T + (3.99e12 - 6.91e12i)T^{2}
43 1+(4.95e5−8.57e5i)T+(−5.84e12−1.01e13i)T2 1 + (4.95e5 - 8.57e5i)T + (-5.84e12 - 1.01e13i)T^{2}
47 1+(−5.80e6−3.35e6i)T+(1.19e13+2.06e13i)T2 1 + (-5.80e6 - 3.35e6i)T + (1.19e13 + 2.06e13i)T^{2}
53 1−1.00e7iT−6.22e13T2 1 - 1.00e7iT - 6.22e13T^{2}
59 1+(−1.38e6+7.99e5i)T+(7.34e13−1.27e14i)T2 1 + (-1.38e6 + 7.99e5i)T + (7.34e13 - 1.27e14i)T^{2}
61 1+(9.68e6−1.67e7i)T+(−9.58e13−1.66e14i)T2 1 + (9.68e6 - 1.67e7i)T + (-9.58e13 - 1.66e14i)T^{2}
67 1+(−1.40e7−2.42e7i)T+(−2.03e14+3.51e14i)T2 1 + (-1.40e7 - 2.42e7i)T + (-2.03e14 + 3.51e14i)T^{2}
71 1+3.36e7iT−6.45e14T2 1 + 3.36e7iT - 6.45e14T^{2}
73 1+2.52e7T+8.06e14T2 1 + 2.52e7T + 8.06e14T^{2}
79 1+(−3.17e7+5.49e7i)T+(−7.58e14−1.31e15i)T2 1 + (-3.17e7 + 5.49e7i)T + (-7.58e14 - 1.31e15i)T^{2}
83 1+(4.11e7+2.37e7i)T+(1.12e15+1.95e15i)T2 1 + (4.11e7 + 2.37e7i)T + (1.12e15 + 1.95e15i)T^{2}
89 1+7.82e7iT−3.93e15T2 1 + 7.82e7iT - 3.93e15T^{2}
97 1+(9.77e6−1.69e7i)T+(−3.91e15−6.78e15i)T2 1 + (9.77e6 - 1.69e7i)T + (-3.91e15 - 6.78e15i)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

−10.31582293240025704686560383782, −8.678287061796805546243948405185, −7.81009913586690114673068511395, −7.24530518010861372501426285287, −6.27135937933247699979368148931, −4.46726885611998399805418049954, −4.01335631690749118353639656327, −2.90853848955852167003406593307, −1.30114900938714044914439889994, −0.082460914062800496435394924326, 0.924070345090790133409683396140, 2.34219289209548342589335809049, 3.68960808543985377799557770690, 4.68287513344033440161377303150, 5.43026579478538334181023058169, 6.87930607289050590409246193557, 8.146469048662545949180545386406, 8.391542997547529727329145217825, 9.365155393731117511918114021805, 10.80336085884998647090786744545

Graph of the ZZ-function along the critical line