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Properties

Label 2-18e2-3.2-c8-0-15
Degree 22
Conductor 324324
Sign −i-i
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  + 878. i·5-s + 1.52e3·7-s − 1.01e4i·11-s − 7.95e3·13-s + 9.94e4i·17-s + 1.05e5·19-s − 2.04e5i·23-s − 3.81e5·25-s + 4.32e5i·29-s + 1.06e6·31-s + 1.33e6i·35-s + 3.38e6·37-s + 2.50e6i·41-s + 6.56e6·43-s − 8.43e6i·47-s + ⋯
L(s)  = 1  + 1.40i·5-s + 0.634·7-s − 0.694i·11-s − 0.278·13-s + 1.19i·17-s + 0.807·19-s − 0.731i·23-s − 0.977·25-s + 0.612i·29-s + 1.15·31-s + 0.892i·35-s + 1.80·37-s + 0.887i·41-s + 1.92·43-s − 1.72i·47-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 324324    =    22⋅342^{2} \cdot 3^{4}
Sign: −i-i
Analytic conductor: 131.990131.990
Root analytic conductor: 11.488711.4887
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ324(161,⋅)\chi_{324} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 324, ( :4), −i)(2,\ 324,\ (\ :4),\ -i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 2.4345235212.434523521
L(12)L(\frac12) ≈\approx 2.4345235212.434523521
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 1−878.iT−3.90e5T2 1 - 878. iT - 3.90e5T^{2}
7 1−1.52e3T+5.76e6T2 1 - 1.52e3T + 5.76e6T^{2}
11 1+1.01e4iT−2.14e8T2 1 + 1.01e4iT - 2.14e8T^{2}
13 1+7.95e3T+8.15e8T2 1 + 7.95e3T + 8.15e8T^{2}
17 1−9.94e4iT−6.97e9T2 1 - 9.94e4iT - 6.97e9T^{2}
19 1−1.05e5T+1.69e10T2 1 - 1.05e5T + 1.69e10T^{2}
23 1+2.04e5iT−7.83e10T2 1 + 2.04e5iT - 7.83e10T^{2}
29 1−4.32e5iT−5.00e11T2 1 - 4.32e5iT - 5.00e11T^{2}
31 1−1.06e6T+8.52e11T2 1 - 1.06e6T + 8.52e11T^{2}
37 1−3.38e6T+3.51e12T2 1 - 3.38e6T + 3.51e12T^{2}
41 1−2.50e6iT−7.98e12T2 1 - 2.50e6iT - 7.98e12T^{2}
43 1−6.56e6T+1.16e13T2 1 - 6.56e6T + 1.16e13T^{2}
47 1+8.43e6iT−2.38e13T2 1 + 8.43e6iT - 2.38e13T^{2}
53 1−1.67e6iT−6.22e13T2 1 - 1.67e6iT - 6.22e13T^{2}
59 1+3.81e6iT−1.46e14T2 1 + 3.81e6iT - 1.46e14T^{2}
61 1+2.51e7T+1.91e14T2 1 + 2.51e7T + 1.91e14T^{2}
67 1+3.27e6T+4.06e14T2 1 + 3.27e6T + 4.06e14T^{2}
71 1−4.56e7iT−6.45e14T2 1 - 4.56e7iT - 6.45e14T^{2}
73 1−4.28e7T+8.06e14T2 1 - 4.28e7T + 8.06e14T^{2}
79 1+3.26e7T+1.51e15T2 1 + 3.26e7T + 1.51e15T^{2}
83 1+7.07e7iT−2.25e15T2 1 + 7.07e7iT - 2.25e15T^{2}
89 1−6.81e7iT−3.93e15T2 1 - 6.81e7iT - 3.93e15T^{2}
97 1+4.47e7T+7.83e15T2 1 + 4.47e7T + 7.83e15T^{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

−10.65534309921497161171784879842, −9.706165799833141813488334657672, −8.411824290668742360516826973017, −7.63625769975655039198025540288, −6.59324146480464332114987833381, −5.80499231712798552535941435758, −4.43238894150370740779154393448, −3.25437230799707099764719368514, −2.38615420900548481163631898704, −1.00589612474288128278077552229, 0.58400016901776535189870221275, 1.40152188287431314168932426022, 2.67996097843447895802829838776, 4.38069689531042784988885059820, 4.87285599448851524929704108528, 5.89187391759015083808221601973, 7.44258074324749514839928157660, 8.009219528407905226953108827636, 9.299833629268881993984740009027, 9.588799337475076191950594882375

Graph of the ZZ-function along the critical line