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Properties

Label 2-82110-1.1-c1-0-34
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 + 3·11-s − 12-s + 2·13-s + 14-s − 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s − 3·22-s + 23-s + 24-s + 25-s − 2·26-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 + 0.904·11-s − 0.288·12-s + 0.554·13-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 − 0.639·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-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 1−3T+pT2 1 - 3 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1−9T+pT2 1 - 9 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+11T+pT2 1 + 11 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+14T+pT2 1 + 14 T + p T^{2}
61 1+9T+pT2 1 + 9 T + p T^{2}
67 1+13T+pT2 1 + 13 T + p T^{2}
71 1−11T+pT2 1 - 11 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1+11T+pT2 1 + 11 T + p T^{2}
97 1+13T+pT2 1 + 13 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

−14.12541673227984, −13.55140515087466, −13.40790657420857, −12.51275149014399, −12.13462247880189, −11.76260861766436, −11.09387419566167, −10.71805320410787, −10.23505736242731, −9.604743967038337, −9.367395497522360, −8.664796981538845, −8.310970267610410, −7.558799504237692, −6.999005054144196, −6.371385483175066, −6.224391332487146, −5.650076650914217, −4.792135563481625, −4.290814165137145, −3.638160182130162, −2.807128574356339, −2.323396680980555, −1.263092767697954, −1.057469327412862, 0, 1.057469327412862, 1.263092767697954, 2.323396680980555, 2.807128574356339, 3.638160182130162, 4.290814165137145, 4.792135563481625, 5.650076650914217, 6.224391332487146, 6.371385483175066, 6.999005054144196, 7.558799504237692, 8.310970267610410, 8.664796981538845, 9.367395497522360, 9.604743967038337, 10.23505736242731, 10.71805320410787, 11.09387419566167, 11.76260861766436, 12.13462247880189, 12.51275149014399, 13.40790657420857, 13.55140515087466, 14.12541673227984

Graph of the ZZ-function along the critical line