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Properties

Label 2-475-1.1-c1-0-23
Degree 22
Conductor 475475
Sign −1-1
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.65·2-s − 2.37·3-s + 0.726·4-s − 3.92·6-s + 0.377·7-s − 2.10·8-s + 2.65·9-s − 1.37·11-s − 1.72·12-s − 2.82·13-s + 0.622·14-s − 4.92·16-s − 6.37·17-s + 4.37·18-s + 19-s − 0.896·21-s − 2.27·22-s − 6.19·23-s + 4.99·24-s − 4.65·26-s + 0.829·27-s + 0.273·28-s − 3.37·29-s + 2.48·31-s − 3.92·32-s + 3.27·33-s − 10.5·34-s + ⋯
L(s)  = 1  + 1.16·2-s − 1.37·3-s + 0.363·4-s − 1.60·6-s + 0.142·7-s − 0.743·8-s + 0.883·9-s − 0.415·11-s − 0.498·12-s − 0.782·13-s + 0.166·14-s − 1.23·16-s − 1.54·17-s + 1.03·18-s + 0.229·19-s − 0.195·21-s − 0.484·22-s − 1.29·23-s + 1.02·24-s − 0.913·26-s + 0.159·27-s + 0.0517·28-s − 0.627·29-s + 0.445·31-s − 0.693·32-s + 0.569·33-s − 1.80·34-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: −1-1
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 475, ( :1/2), −1)(2,\ 475,\ (\ :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)
bad5 1 1
19 1−T 1 - T
good2 1−1.65T+2T2 1 - 1.65T + 2T^{2}
3 1+2.37T+3T2 1 + 2.37T + 3T^{2}
7 1−0.377T+7T2 1 - 0.377T + 7T^{2}
11 1+1.37T+11T2 1 + 1.37T + 11T^{2}
13 1+2.82T+13T2 1 + 2.82T + 13T^{2}
17 1+6.37T+17T2 1 + 6.37T + 17T^{2}
23 1+6.19T+23T2 1 + 6.19T + 23T^{2}
29 1+3.37T+29T2 1 + 3.37T + 29T^{2}
31 1−2.48T+31T2 1 - 2.48T + 31T^{2}
37 1+5.58T+37T2 1 + 5.58T + 37T^{2}
41 1−8.50T+41T2 1 - 8.50T + 41T^{2}
43 1−12.1T+43T2 1 - 12.1T + 43T^{2}
47 1−6.87T+47T2 1 - 6.87T + 47T^{2}
53 1+11.5T+53T2 1 + 11.5T + 53T^{2}
59 1−6.05T+59T2 1 - 6.05T + 59T^{2}
61 1−5.02T+61T2 1 - 5.02T + 61T^{2}
67 1+3.22T+67T2 1 + 3.22T + 67T^{2}
71 1+2.30T+71T2 1 + 2.30T + 71T^{2}
73 1−3.19T+73T2 1 - 3.19T + 73T^{2}
79 1+6.71T+79T2 1 + 6.71T + 79T^{2}
83 1+18.2T+83T2 1 + 18.2T + 83T^{2}
89 1−1.50T+89T2 1 - 1.50T + 89T^{2}
97 1−11.7T+97T2 1 - 11.7T + 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

−10.99557923397817890835730887741, −9.929999269860691039043409127078, −8.840502704715681938073667789669, −7.43647634996632542907942242136, −6.37029850032284263600142052056, −5.70185668950312104610222386817, −4.83824677001158109884965668823, −4.12650126542128489161939162915, −2.47046698591722202765913608213, 0, 2.47046698591722202765913608213, 4.12650126542128489161939162915, 4.83824677001158109884965668823, 5.70185668950312104610222386817, 6.37029850032284263600142052056, 7.43647634996632542907942242136, 8.840502704715681938073667789669, 9.929999269860691039043409127078, 10.99557923397817890835730887741

Graph of the ZZ-function along the critical line