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Properties

Label 2-82110-1.1-c1-0-51
Degree 22
Conductor 8211082110
Sign 11
Analytic cond. 655.651655.651
Root an. cond. 25.605625.6056
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 6·11-s + 12-s − 4·13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 4·19-s + 20-s + 21-s + 6·22-s − 23-s − 24-s + 25-s + 4·26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s + 0.288·12-s − 1.10·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s + 1.27·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(82110s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(82110s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8211082110    =    2⋅3⋅5⋅7⋅17⋅232 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23
Sign: 11
Analytic conductor: 655.651655.651
Root analytic conductor: 25.605625.6056
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (2, 82110, ( :1/2), 1)(2,\ 82110,\ (\ :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+T 1 + T
3 1−T 1 - T
5 1−T 1 - T
7 1−T 1 - T
17 1+T 1 + T
23 1+T 1 + T
good11 1+6T+pT2 1 + 6 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+10T+pT2 1 + 10 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−14.78844437531077, −13.89008958876934, −13.44332517720600, −13.00667223176805, −12.52539764533131, −12.08210825574281, −11.26532070112769, −10.74722534290345, −10.48960480943400, −9.956664215173367, −9.374525705072295, −8.991689130419675, −8.410541121729538, −7.842680761400633, −7.456657927453526, −7.144755999421428, −6.269149202595982, −5.668203716424800, −5.169694767761840, −4.608288234659486, −3.892970429262524, −2.938586461622653, −2.652532499514128, −1.899979904608380, −1.599222173373442, 0, 0, 1.599222173373442, 1.899979904608380, 2.652532499514128, 2.938586461622653, 3.892970429262524, 4.608288234659486, 5.169694767761840, 5.668203716424800, 6.269149202595982, 7.144755999421428, 7.456657927453526, 7.842680761400633, 8.410541121729538, 8.991689130419675, 9.374525705072295, 9.956664215173367, 10.48960480943400, 10.74722534290345, 11.26532070112769, 12.08210825574281, 12.52539764533131, 13.00667223176805, 13.44332517720600, 13.89008958876934, 14.78844437531077

Graph of the ZZ-function along the critical line