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Properties

Label 2-82110-1.1-c1-0-21
Degree 22
Conductor 8211082110
Sign −1-1
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 11

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 − 2·11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s + 20-s + 21-s + 2·22-s − 23-s + 24-s + 25-s + 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 − 0.603·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-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: −1-1
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: 11
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+2T+pT2 1 + 2 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−5T+pT2 1 - 5 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1−6T+pT2 1 - 6 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

−14.20802939572624, −13.67003949764397, −13.13387009240808, −12.59352528290151, −12.28252588445129, −11.66712354688956, −11.03414230916866, −10.78343562800785, −10.08770144108048, −9.765514031101350, −9.405620752412463, −8.677608175926759, −8.048909864647366, −7.750166977044103, −7.012013193862558, −6.457596520300363, −6.162356107306313, −5.452056615953185, −4.986730045983116, −4.303065143829011, −3.577585466708620, −2.701334084647574, −2.415956289644347, −1.472095977663463, −0.8064877912286052, 0, 0.8064877912286052, 1.472095977663463, 2.415956289644347, 2.701334084647574, 3.577585466708620, 4.303065143829011, 4.986730045983116, 5.452056615953185, 6.162356107306313, 6.457596520300363, 7.012013193862558, 7.750166977044103, 8.048909864647366, 8.677608175926759, 9.405620752412463, 9.765514031101350, 10.08770144108048, 10.78343562800785, 11.03414230916866, 11.66712354688956, 12.28252588445129, 12.59352528290151, 13.13387009240808, 13.67003949764397, 14.20802939572624

Graph of the ZZ-function along the critical line