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Properties

Label 2-5225-1.1-c1-0-96
Degree 22
Conductor 52255225
Sign 11
Analytic cond. 41.721841.7218
Root an. cond. 6.459246.45924
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.527·2-s + 1.24·3-s − 1.72·4-s − 0.656·6-s + 3.22·7-s + 1.96·8-s − 1.45·9-s + 11-s − 2.14·12-s − 4.87·13-s − 1.69·14-s + 2.41·16-s + 7.11·17-s + 0.764·18-s + 19-s + 4.01·21-s − 0.527·22-s + 1.12·23-s + 2.44·24-s + 2.56·26-s − 5.53·27-s − 5.54·28-s + 8.47·29-s − 0.926·31-s − 5.19·32-s + 1.24·33-s − 3.75·34-s + ⋯
L(s)  = 1  − 0.372·2-s + 0.718·3-s − 0.861·4-s − 0.267·6-s + 1.21·7-s + 0.693·8-s − 0.483·9-s + 0.301·11-s − 0.618·12-s − 1.35·13-s − 0.453·14-s + 0.602·16-s + 1.72·17-s + 0.180·18-s + 0.229·19-s + 0.875·21-s − 0.112·22-s + 0.234·23-s + 0.498·24-s + 0.503·26-s − 1.06·27-s − 1.04·28-s + 1.57·29-s − 0.166·31-s − 0.918·32-s + 0.216·33-s − 0.643·34-s + ⋯

Functional equation

Λ(s)=(5225s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5225s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 52255225    =    52⋅11⋅195^{2} \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 41.721841.7218
Root analytic conductor: 6.459246.45924
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5225, ( :1/2), 1)(2,\ 5225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8704359381.870435938
L(12)L(\frac12) ≈\approx 1.8704359381.870435938
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)
bad5 1 1
11 1−T 1 - T
19 1−T 1 - T
good2 1+0.527T+2T2 1 + 0.527T + 2T^{2}
3 1−1.24T+3T2 1 - 1.24T + 3T^{2}
7 1−3.22T+7T2 1 - 3.22T + 7T^{2}
13 1+4.87T+13T2 1 + 4.87T + 13T^{2}
17 1−7.11T+17T2 1 - 7.11T + 17T^{2}
23 1−1.12T+23T2 1 - 1.12T + 23T^{2}
29 1−8.47T+29T2 1 - 8.47T + 29T^{2}
31 1+0.926T+31T2 1 + 0.926T + 31T^{2}
37 1+10.7T+37T2 1 + 10.7T + 37T^{2}
41 1−7.49T+41T2 1 - 7.49T + 41T^{2}
43 1+7.22T+43T2 1 + 7.22T + 43T^{2}
47 1−12.4T+47T2 1 - 12.4T + 47T^{2}
53 1−7.92T+53T2 1 - 7.92T + 53T^{2}
59 1−1.91T+59T2 1 - 1.91T + 59T^{2}
61 1+11.5T+61T2 1 + 11.5T + 61T^{2}
67 1−10.2T+67T2 1 - 10.2T + 67T^{2}
71 1−0.979T+71T2 1 - 0.979T + 71T^{2}
73 1+15.3T+73T2 1 + 15.3T + 73T^{2}
79 1+3.05T+79T2 1 + 3.05T + 79T^{2}
83 1+2.60T+83T2 1 + 2.60T + 83T^{2}
89 1−5.38T+89T2 1 - 5.38T + 89T^{2}
97 1−15.6T+97T2 1 - 15.6T + 97T^{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

−8.338665383136570472608961200485, −7.60202564589714069484662642263, −7.28130744178078441068854334425, −5.81452308333526169235393053542, −5.15434450573115415249866931915, −4.62720023883456843208842850415, −3.66184503182185092303404418753, −2.84540344504849066589766071853, −1.80492518997956083140819796008, −0.791238087005524469002005564743, 0.791238087005524469002005564743, 1.80492518997956083140819796008, 2.84540344504849066589766071853, 3.66184503182185092303404418753, 4.62720023883456843208842850415, 5.15434450573115415249866931915, 5.81452308333526169235393053542, 7.28130744178078441068854334425, 7.60202564589714069484662642263, 8.338665383136570472608961200485

Graph of the ZZ-function along the critical line