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Properties

Label 2-832-1.1-c5-0-22
Degree 22
Conductor 832832
Sign 11
Analytic cond. 133.439133.439
Root an. cond. 11.551511.5515
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·5-s − 170·7-s − 243·9-s + 250·11-s + 169·13-s + 1.06e3·17-s + 78·19-s + 1.57e3·23-s − 2.92e3·25-s − 2.57e3·29-s − 8.65e3·31-s − 2.38e3·35-s − 1.09e4·37-s + 1.05e3·41-s + 5.90e3·43-s − 3.40e3·45-s − 5.96e3·47-s + 1.20e4·49-s − 2.90e4·53-s + 3.50e3·55-s + 1.39e4·59-s + 3.28e4·61-s + 4.13e4·63-s + 2.36e3·65-s + 6.95e4·67-s − 5.05e4·71-s − 4.67e4·73-s + ⋯
L(s)  = 1  + 0.250·5-s − 1.31·7-s − 9-s + 0.622·11-s + 0.277·13-s + 0.891·17-s + 0.0495·19-s + 0.621·23-s − 0.937·25-s − 0.569·29-s − 1.61·31-s − 0.328·35-s − 1.31·37-s + 0.0975·41-s + 0.486·43-s − 0.250·45-s − 0.393·47-s + 0.719·49-s − 1.42·53-s + 0.156·55-s + 0.520·59-s + 1.13·61-s + 1.31·63-s + 0.0694·65-s + 1.89·67-s − 1.18·71-s − 1.02·73-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 11
Analytic conductor: 133.439133.439
Root analytic conductor: 11.551511.5515
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 832, ( :5/2), 1)(2,\ 832,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.3183894181.318389418
L(12)L(\frac12) ≈\approx 1.3183894181.318389418
L(72)L(\frac{7}{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
13 1−p2T 1 - p^{2} T
good3 1+p5T2 1 + p^{5} T^{2}
5 1−14T+p5T2 1 - 14 T + p^{5} T^{2}
7 1+170T+p5T2 1 + 170 T + p^{5} T^{2}
11 1−250T+p5T2 1 - 250 T + p^{5} T^{2}
17 1−1062T+p5T2 1 - 1062 T + p^{5} T^{2}
19 1−78T+p5T2 1 - 78 T + p^{5} T^{2}
23 1−1576T+p5T2 1 - 1576 T + p^{5} T^{2}
29 1+2578T+p5T2 1 + 2578 T + p^{5} T^{2}
31 1+8654T+p5T2 1 + 8654 T + p^{5} T^{2}
37 1+10986T+p5T2 1 + 10986 T + p^{5} T^{2}
41 1−1050T+p5T2 1 - 1050 T + p^{5} T^{2}
43 1−5900T+p5T2 1 - 5900 T + p^{5} T^{2}
47 1+5962T+p5T2 1 + 5962 T + p^{5} T^{2}
53 1+29046T+p5T2 1 + 29046 T + p^{5} T^{2}
59 1−13922T+p5T2 1 - 13922 T + p^{5} T^{2}
61 1−32882T+p5T2 1 - 32882 T + p^{5} T^{2}
67 1−69566T+p5T2 1 - 69566 T + p^{5} T^{2}
71 1+50542T+p5T2 1 + 50542 T + p^{5} T^{2}
73 1+46750T+p5T2 1 + 46750 T + p^{5} T^{2}
79 1+19348T+p5T2 1 + 19348 T + p^{5} T^{2}
83 1−87438T+p5T2 1 - 87438 T + p^{5} T^{2}
89 1−94170T+p5T2 1 - 94170 T + p^{5} T^{2}
97 1−182786T+p5T2 1 - 182786 T + p^{5} 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

−9.377229853524475265579025959374, −8.868530185821819237018558081424, −7.72751070452978243477673560153, −6.76000859915714781081870400028, −5.97215373829448459898122953185, −5.30072386428473905638344010075, −3.70036652818864040850115564705, −3.20769602123505276858984146027, −1.90440122665330166531145249177, −0.50343746788355741463112735588, 0.50343746788355741463112735588, 1.90440122665330166531145249177, 3.20769602123505276858984146027, 3.70036652818864040850115564705, 5.30072386428473905638344010075, 5.97215373829448459898122953185, 6.76000859915714781081870400028, 7.72751070452978243477673560153, 8.868530185821819237018558081424, 9.377229853524475265579025959374

Graph of the ZZ-function along the critical line