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Properties

Label 2-2175-1.1-c1-0-17
Degree 22
Conductor 21752175
Sign 11
Analytic cond. 17.367417.3674
Root an. cond. 4.167424.16742
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  − 2·2-s + 3-s + 2·4-s − 2·6-s − 2·7-s + 9-s − 3·11-s + 2·12-s + 4·13-s + 4·14-s − 4·16-s + 8·17-s − 2·18-s − 2·21-s + 6·22-s − 23-s − 8·26-s + 27-s − 4·28-s + 29-s − 8·31-s + 8·32-s − 3·33-s − 16·34-s + 2·36-s − 7·37-s + 4·39-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 0.755·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s + 1.10·13-s + 1.06·14-s − 16-s + 1.94·17-s − 0.471·18-s − 0.436·21-s + 1.27·22-s − 0.208·23-s − 1.56·26-s + 0.192·27-s − 0.755·28-s + 0.185·29-s − 1.43·31-s + 1.41·32-s − 0.522·33-s − 2.74·34-s + 1/3·36-s − 1.15·37-s + 0.640·39-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: 11
Analytic conductor: 17.367417.3674
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2175, ( :1/2), 1)(2,\ 2175,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.92871798090.9287179809
L(12)L(\frac12) ≈\approx 0.92871798090.9287179809
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)
bad3 1−T 1 - T
5 1 1
29 1−T 1 - T
good2 1+pT+pT2 1 + p T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−8T+pT2 1 - 8 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+T+pT2 1 + T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1−7T+pT2 1 - 7 T + p T^{2}
43 1−9T+pT2 1 - 9 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+T+pT2 1 + T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−9T+pT2 1 - 9 T + p T^{2}
89 1−10T+pT2 1 - 10 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

−9.026204258303570972775141705216, −8.393055517263135127517562818868, −7.71321986683429932357831257897, −7.18292635323516030785900459786, −6.12204212796184096175443912425, −5.25685379924244030560862286764, −3.83993387725300169108109456622, −3.09220257677483752161515538695, −1.91602304596019401141580620431, −0.76989224128822694234011319404, 0.76989224128822694234011319404, 1.91602304596019401141580620431, 3.09220257677483752161515538695, 3.83993387725300169108109456622, 5.25685379924244030560862286764, 6.12204212796184096175443912425, 7.18292635323516030785900459786, 7.71321986683429932357831257897, 8.393055517263135127517562818868, 9.026204258303570972775141705216

Graph of the ZZ-function along the critical line