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Properties

Label 2-22050-1.1-c1-0-12
Degree 22
Conductor 2205022050
Sign 11
Analytic cond. 176.070176.070
Root an. cond. 13.269113.2691
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 + 4-s − 8-s − 2·13-s + 16-s − 3·17-s + 8·19-s − 9·23-s + 2·26-s + 6·29-s + 5·31-s − 32-s + 3·34-s − 8·37-s − 8·38-s + 3·41-s + 10·43-s + 9·46-s − 3·47-s − 2·52-s + 6·53-s − 6·58-s − 12·59-s − 4·61-s − 5·62-s + 64-s − 2·67-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.554·13-s + 1/4·16-s − 0.727·17-s + 1.83·19-s − 1.87·23-s + 0.392·26-s + 1.11·29-s + 0.898·31-s − 0.176·32-s + 0.514·34-s − 1.31·37-s − 1.29·38-s + 0.468·41-s + 1.52·43-s + 1.32·46-s − 0.437·47-s − 0.277·52-s + 0.824·53-s − 0.787·58-s − 1.56·59-s − 0.512·61-s − 0.635·62-s + 1/8·64-s − 0.244·67-s + ⋯

Functional equation

Λ(s)=(22050s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(22050s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2205022050    =    2⋅32⋅52⋅722 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 176.070176.070
Root analytic conductor: 13.269113.2691
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 22050, ( :1/2), 1)(2,\ 22050,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2526377021.252637702
L(12)L(\frac12) ≈\approx 1.2526377021.252637702
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 1
5 1 1
7 1 1
good11 1+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+9T+pT2 1 + 9 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−9T+pT2 1 - 9 T + p T^{2}
73 1−10T+pT2 1 - 10 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+3T+pT2 1 + 3 T + p T^{2}
97 1+5T+pT2 1 + 5 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

−15.68822174695765, −15.27143579227101, −14.29221575193003, −13.93219977381196, −13.62849353993297, −12.53862950846915, −12.15060184718702, −11.79971683252454, −11.06303790282702, −10.49925352385125, −9.903018250847937, −9.531104503176265, −8.908407666398577, −8.193592466272724, −7.761166255072556, −7.176886496417561, −6.521287722592530, −5.925357702764580, −5.220919918664352, −4.516266511175670, −3.751537038047441, −2.906520907422095, −2.305889809209739, −1.431358313395624, −0.5222581308184795, 0.5222581308184795, 1.431358313395624, 2.305889809209739, 2.906520907422095, 3.751537038047441, 4.516266511175670, 5.220919918664352, 5.925357702764580, 6.521287722592530, 7.176886496417561, 7.761166255072556, 8.193592466272724, 8.908407666398577, 9.531104503176265, 9.903018250847937, 10.49925352385125, 11.06303790282702, 11.79971683252454, 12.15060184718702, 12.53862950846915, 13.62849353993297, 13.93219977381196, 14.29221575193003, 15.27143579227101, 15.68822174695765

Graph of the ZZ-function along the critical line