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Properties

Label 2-82110-1.1-c1-0-1
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 00

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 − 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 5·19-s − 20-s + 21-s + 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 − 0.301·11-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 + 1.14·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-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: 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: 00
Selberg data: (2, 82110, ( :1/2), 1)(2,\ 82110,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.31263483780.3126348378
L(12)L(\frac12) ≈\approx 0.31263483780.3126348378
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+T+pT2 1 + T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+7T+pT2 1 + 7 T + p T^{2}
53 1+T+pT2 1 + T + p T^{2}
59 1−7T+pT2 1 - 7 T + p T^{2}
61 1+15T+pT2 1 + 15 T + p T^{2}
67 1+T+pT2 1 + T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+6T+pT2 1 + 6 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−3T+pT2 1 - 3 T + p T^{2}
97 1+7T+pT2 1 + 7 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

−13.93356675399703, −13.39746323979951, −12.88321334423791, −12.29758056734258, −11.97764759426959, −11.34436637341654, −11.01438126067602, −10.48561028161175, −10.00943644731680, −9.436815774983529, −8.987811861501665, −8.449882631006038, −7.796730453067150, −7.355803530554789, −6.878574201791485, −6.406049582268974, −5.618133758067181, −5.299848153297241, −4.627457635830215, −3.746105102078350, −3.381432656919009, −2.664949235627336, −1.761220712409575, −1.207554275468046, −0.2251894697401192, 0.2251894697401192, 1.207554275468046, 1.761220712409575, 2.664949235627336, 3.381432656919009, 3.746105102078350, 4.627457635830215, 5.299848153297241, 5.618133758067181, 6.406049582268974, 6.878574201791485, 7.355803530554789, 7.796730453067150, 8.449882631006038, 8.987811861501665, 9.436815774983529, 10.00943644731680, 10.48561028161175, 11.01438126067602, 11.34436637341654, 11.97764759426959, 12.29758056734258, 12.88321334423791, 13.39746323979951, 13.93356675399703

Graph of the ZZ-function along the critical line