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Properties

Label 2-4032-1.1-c1-0-47
Degree 22
Conductor 40324032
Sign −1-1
Analytic cond. 32.195632.1956
Root an. cond. 5.674125.67412
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 6·11-s − 2·13-s − 4·19-s − 6·23-s − 5·25-s + 6·29-s − 8·31-s − 2·37-s − 12·41-s − 4·43-s + 12·47-s + 49-s − 6·53-s + 10·61-s + 8·67-s + 6·71-s − 10·73-s − 6·77-s + 4·79-s + 12·83-s − 12·89-s + 2·91-s − 10·97-s − 12·101-s − 8·103-s + 6·107-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.80·11-s − 0.554·13-s − 0.917·19-s − 1.25·23-s − 25-s + 1.11·29-s − 1.43·31-s − 0.328·37-s − 1.87·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s + 1.28·61-s + 0.977·67-s + 0.712·71-s − 1.17·73-s − 0.683·77-s + 0.450·79-s + 1.31·83-s − 1.27·89-s + 0.209·91-s − 1.01·97-s − 1.19·101-s − 0.788·103-s + 0.580·107-s + ⋯

Functional equation

Λ(s)=(4032s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4032s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40324032    =    26⋅32⋅72^{6} \cdot 3^{2} \cdot 7
Sign: −1-1
Analytic conductor: 32.195632.1956
Root analytic conductor: 5.674125.67412
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4032, ( :1/2), −1)(2,\ 4032,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
7 1+T 1 + T
good5 1+pT2 1 + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p 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

−8.213446609484277420706681145366, −7.18052525439954131151018372494, −6.61214442273728283164556360378, −6.01676531608062216159532650200, −5.07643921282730531606229802841, −3.99784334743039531176458960720, −3.70585619782695729246427373887, −2.36003185284708036230417844213, −1.49990868331367069680950997074, 0, 1.49990868331367069680950997074, 2.36003185284708036230417844213, 3.70585619782695729246427373887, 3.99784334743039531176458960720, 5.07643921282730531606229802841, 6.01676531608062216159532650200, 6.61214442273728283164556360378, 7.18052525439954131151018372494, 8.213446609484277420706681145366

Graph of the ZZ-function along the critical line