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Properties

Label 2-5610-1.1-c1-0-21
Degree 22
Conductor 56105610
Sign 11
Analytic cond. 44.796044.7960
Root an. cond. 6.692986.69298
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 11-s − 12-s + 2·13-s − 15-s + 16-s − 17-s − 18-s + 4·19-s + 20-s + 22-s + 8·23-s + 24-s + 25-s − 2·26-s − 27-s + 6·29-s + 30-s + 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.554·13-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 + 0.213·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.718·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(5610s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5610s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 56105610    =    2⋅3⋅5⋅11⋅172 \cdot 3 \cdot 5 \cdot 11 \cdot 17
Sign: 11
Analytic conductor: 44.796044.7960
Root analytic conductor: 6.692986.69298
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5610, ( :1/2), 1)(2,\ 5610,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3318864091.331886409
L(12)L(\frac12) ≈\approx 1.3318864091.331886409
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
11 1+T 1 + T
17 1+T 1 + T
good7 1+pT2 1 + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−2T+pT2 1 - 2 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

−8.263245916652137648768056548346, −7.35284289134810156595966275878, −6.73034022983676842963480004857, −6.17979730105811009829056357854, −5.24782021011330946478114343156, −4.78445616198338088336599939909, −3.47722324865831194243662083729, −2.72558452751296671636591477988, −1.57147525857787416140079301837, −0.75049248480518745009477310682, 0.75049248480518745009477310682, 1.57147525857787416140079301837, 2.72558452751296671636591477988, 3.47722324865831194243662083729, 4.78445616198338088336599939909, 5.24782021011330946478114343156, 6.17979730105811009829056357854, 6.73034022983676842963480004857, 7.35284289134810156595966275878, 8.263245916652137648768056548346

Graph of the ZZ-function along the critical line