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Properties

Label 2-2592-3.2-c2-0-3
Degree 22
Conductor 25922592
Sign −1-1
Analytic cond. 70.626870.6268
Root an. cond. 8.403988.40398
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.832i·5-s + 2.52·7-s + 10.9i·11-s − 8.72·13-s + 20.8i·17-s − 1.50·19-s + 1.15i·23-s + 24.3·25-s − 18.1i·29-s − 51.3·31-s − 2.09i·35-s − 7.93·37-s − 25.2i·41-s + 38.6·43-s + 68.8i·47-s + ⋯
L(s)  = 1  − 0.166i·5-s + 0.360·7-s + 0.994i·11-s − 0.671·13-s + 1.22i·17-s − 0.0790·19-s + 0.0503i·23-s + 0.972·25-s − 0.626i·29-s − 1.65·31-s − 0.0599i·35-s − 0.214·37-s − 0.616i·41-s + 0.899·43-s + 1.46i·47-s + ⋯

Functional equation

Λ(s)=(2592s/2 ΓC(s) L(s)=(− Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2592s/2 ΓC(s+1) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25922592    =    25⋅342^{5} \cdot 3^{4}
Sign: −1-1
Analytic conductor: 70.626870.6268
Root analytic conductor: 8.403988.40398
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2592(161,⋅)\chi_{2592} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2592, ( :1), −1)(2,\ 2592,\ (\ :1),\ -1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.27972217890.2797221789
L(12)L(\frac12) ≈\approx 0.27972217890.2797221789
L(2)L(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 1
3 1 1
good5 1+0.832iT−25T2 1 + 0.832iT - 25T^{2}
7 1−2.52T+49T2 1 - 2.52T + 49T^{2}
11 1−10.9iT−121T2 1 - 10.9iT - 121T^{2}
13 1+8.72T+169T2 1 + 8.72T + 169T^{2}
17 1−20.8iT−289T2 1 - 20.8iT - 289T^{2}
19 1+1.50T+361T2 1 + 1.50T + 361T^{2}
23 1−1.15iT−529T2 1 - 1.15iT - 529T^{2}
29 1+18.1iT−841T2 1 + 18.1iT - 841T^{2}
31 1+51.3T+961T2 1 + 51.3T + 961T^{2}
37 1+7.93T+1.36e3T2 1 + 7.93T + 1.36e3T^{2}
41 1+25.2iT−1.68e3T2 1 + 25.2iT - 1.68e3T^{2}
43 1−38.6T+1.84e3T2 1 - 38.6T + 1.84e3T^{2}
47 1−68.8iT−2.20e3T2 1 - 68.8iT - 2.20e3T^{2}
53 1+46.5iT−2.80e3T2 1 + 46.5iT - 2.80e3T^{2}
59 1+102.iT−3.48e3T2 1 + 102. iT - 3.48e3T^{2}
61 1+88.3T+3.72e3T2 1 + 88.3T + 3.72e3T^{2}
67 1+22.6T+4.48e3T2 1 + 22.6T + 4.48e3T^{2}
71 1+104.iT−5.04e3T2 1 + 104. iT - 5.04e3T^{2}
73 1+75.2T+5.32e3T2 1 + 75.2T + 5.32e3T^{2}
79 1−103.T+6.24e3T2 1 - 103.T + 6.24e3T^{2}
83 1−62.0iT−6.88e3T2 1 - 62.0iT - 6.88e3T^{2}
89 1−1.95iT−7.92e3T2 1 - 1.95iT - 7.92e3T^{2}
97 1+118.T+9.40e3T2 1 + 118.T + 9.40e3T^{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

−9.211265252017805586840193072430, −8.214404685847028919743143788437, −7.62515714680995720704014669152, −6.86173768810247844913682357071, −5.99980059136094838986967626430, −5.06342126662355035175387943881, −4.44212113286540848003998368430, −3.50625426613519406456793072506, −2.27676870552868203902573217232, −1.49780312972647529081046037878, 0.06434818735268672757909301150, 1.28995232984659540889772957903, 2.58611050954368057664360660493, 3.28571318575708496882507668006, 4.41493322603006516038952621437, 5.22413325288899517902175070887, 5.88796779794047603359779058836, 7.01702668276124018503603047220, 7.39127383978954496838946267359, 8.424935950817085136211634069890

Graph of the ZZ-function along the critical line