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Properties

Label 2-18e2-12.11-c3-0-29
Degree 22
Conductor 324324
Sign −0.568−0.822i-0.568 - 0.822i
Analytic cond. 19.116619.1166
Root an. cond. 4.372254.37225
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.50 + 1.31i)2-s + (4.54 + 6.58i)4-s + 19.4i·5-s − 17.4i·7-s + (2.73 + 22.4i)8-s + (−25.5 + 48.6i)10-s + 65.7·11-s − 2.26·13-s + (22.9 − 43.7i)14-s + (−22.6 + 59.8i)16-s + 78.1i·17-s + 15.9i·19-s + (−127. + 88.2i)20-s + (164. + 86.3i)22-s − 180.·23-s + ⋯
L(s)  = 1  + (0.885 + 0.464i)2-s + (0.568 + 0.822i)4-s + 1.73i·5-s − 0.943i·7-s + (0.121 + 0.992i)8-s + (−0.806 + 1.53i)10-s + 1.80·11-s − 0.0483·13-s + (0.438 − 0.835i)14-s + (−0.353 + 0.935i)16-s + 1.11i·17-s + 0.192i·19-s + (−1.42 + 0.986i)20-s + (1.59 + 0.836i)22-s − 1.63·23-s + ⋯

Functional equation

Λ(s)=(324s/2 ΓC(s) L(s)=((−0.568−0.822i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(324s/2 ΓC(s+3/2) L(s)=((−0.568−0.822i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 324324    =    22⋅342^{2} \cdot 3^{4}
Sign: −0.568−0.822i-0.568 - 0.822i
Analytic conductor: 19.116619.1166
Root analytic conductor: 4.372254.37225
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ324(323,⋅)\chi_{324} (323, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 324, ( :3/2), −0.568−0.822i)(2,\ 324,\ (\ :3/2),\ -0.568 - 0.822i)

Particular Values

L(2)L(2) ≈\approx 3.2238267083.223826708
L(12)L(\frac12) ≈\approx 3.2238267083.223826708
L(52)L(\frac{5}{2}) 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+(−2.50−1.31i)T 1 + (-2.50 - 1.31i)T
3 1 1
good5 1−19.4iT−125T2 1 - 19.4iT - 125T^{2}
7 1+17.4iT−343T2 1 + 17.4iT - 343T^{2}
11 1−65.7T+1.33e3T2 1 - 65.7T + 1.33e3T^{2}
13 1+2.26T+2.19e3T2 1 + 2.26T + 2.19e3T^{2}
17 1−78.1iT−4.91e3T2 1 - 78.1iT - 4.91e3T^{2}
19 1−15.9iT−6.85e3T2 1 - 15.9iT - 6.85e3T^{2}
23 1+180.T+1.21e4T2 1 + 180.T + 1.21e4T^{2}
29 1+11.6iT−2.43e4T2 1 + 11.6iT - 2.43e4T^{2}
31 1−30.2iT−2.97e4T2 1 - 30.2iT - 2.97e4T^{2}
37 1+44.8T+5.06e4T2 1 + 44.8T + 5.06e4T^{2}
41 1+307.iT−6.89e4T2 1 + 307. iT - 6.89e4T^{2}
43 1+88.3iT−7.95e4T2 1 + 88.3iT - 7.95e4T^{2}
47 1+44.8T+1.03e5T2 1 + 44.8T + 1.03e5T^{2}
53 1+90.6iT−1.48e5T2 1 + 90.6iT - 1.48e5T^{2}
59 1−605.T+2.05e5T2 1 - 605.T + 2.05e5T^{2}
61 1−283.T+2.26e5T2 1 - 283.T + 2.26e5T^{2}
67 1−622.iT−3.00e5T2 1 - 622. iT - 3.00e5T^{2}
71 1−828.T+3.57e5T2 1 - 828.T + 3.57e5T^{2}
73 1−706.T+3.89e5T2 1 - 706.T + 3.89e5T^{2}
79 1+260.iT−4.93e5T2 1 + 260. iT - 4.93e5T^{2}
83 1−902.T+5.71e5T2 1 - 902.T + 5.71e5T^{2}
89 1+44.4iT−7.04e5T2 1 + 44.4iT - 7.04e5T^{2}
97 1+1.35e3T+9.12e5T2 1 + 1.35e3T + 9.12e5T^{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

−11.54900392184299358859783382148, −10.75167608030434749665766710961, −9.935684251679867203437328576259, −8.334364430202129464788913259302, −7.22904453617433941972808643594, −6.65731275824638003759389454840, −5.92308089604453892603369153358, −3.91471086925480870420171017864, −3.72505577887812501186400757357, −2.04395734371884224611234120503, 0.901732298716070988181601361045, 2.07698336491174386060740949314, 3.82240389376622998754421472842, 4.74294671572722287671498687026, 5.61876883228196295215332081682, 6.58640767252330437596599889829, 8.203843639496537739514929395936, 9.298748381971074523225366509642, 9.619846738170060793126455641234, 11.40736641812087402389072417492

Graph of the ZZ-function along the critical line