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Properties

Label 2-10-1.1-c21-0-1
Degree 22
Conductor 1010
Sign 11
Analytic cond. 27.947727.9477
Root an. cond. 5.286565.28656
Motivic weight 2121
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.02e3·2-s − 1.14e5·3-s + 1.04e6·4-s − 9.76e6·5-s − 1.17e8·6-s − 2.03e8·7-s + 1.07e9·8-s + 2.72e9·9-s − 1.00e10·10-s − 4.40e10·11-s − 1.20e11·12-s − 1.80e11·13-s − 2.08e11·14-s + 1.12e12·15-s + 1.09e12·16-s + 1.29e13·17-s + 2.79e12·18-s + 3.50e13·19-s − 1.02e13·20-s + 2.34e13·21-s − 4.50e13·22-s − 1.00e14·23-s − 1.23e14·24-s + 9.53e13·25-s − 1.85e14·26-s + 8.87e14·27-s − 2.13e14·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.12·3-s + 0.5·4-s − 0.447·5-s − 0.794·6-s − 0.272·7-s + 0.353·8-s + 0.260·9-s − 0.316·10-s − 0.511·11-s − 0.561·12-s − 0.363·13-s − 0.192·14-s + 0.502·15-s + 0.250·16-s + 1.55·17-s + 0.184·18-s + 1.31·19-s − 0.223·20-s + 0.306·21-s − 0.361·22-s − 0.505·23-s − 0.397·24-s + 0.199·25-s − 0.257·26-s + 0.829·27-s − 0.136·28-s + ⋯

Functional equation

Λ(s)=(10s/2 ΓC(s) L(s)=( Λ(22−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}
Λ(s)=(10s/2 ΓC(s+21/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1010    =    2⋅52 \cdot 5
Sign: 11
Analytic conductor: 27.947727.9477
Root analytic conductor: 5.286565.28656
Motivic weight: 2121
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 10, ( :21/2), 1)(2,\ 10,\ (\ :21/2),\ 1)

Particular Values

L(11)L(11) ≈\approx 1.6817587401.681758740
L(12)L(\frac12) ≈\approx 1.6817587401.681758740
L(232)L(\frac{23}{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 1−1.02e3T 1 - 1.02e3T
5 1+9.76e6T 1 + 9.76e6T
good3 1+1.14e5T+1.04e10T2 1 + 1.14e5T + 1.04e10T^{2}
7 1+2.03e8T+5.58e17T2 1 + 2.03e8T + 5.58e17T^{2}
11 1+4.40e10T+7.40e21T2 1 + 4.40e10T + 7.40e21T^{2}
13 1+1.80e11T+2.47e23T2 1 + 1.80e11T + 2.47e23T^{2}
17 1−1.29e13T+6.90e25T2 1 - 1.29e13T + 6.90e25T^{2}
19 1−3.50e13T+7.14e26T2 1 - 3.50e13T + 7.14e26T^{2}
23 1+1.00e14T+3.94e28T2 1 + 1.00e14T + 3.94e28T^{2}
29 1−2.65e15T+5.13e30T2 1 - 2.65e15T + 5.13e30T^{2}
31 1−1.12e15T+2.08e31T2 1 - 1.12e15T + 2.08e31T^{2}
37 1−2.62e16T+8.55e32T2 1 - 2.62e16T + 8.55e32T^{2}
41 1+2.80e16T+7.38e33T2 1 + 2.80e16T + 7.38e33T^{2}
43 1−1.48e17T+2.00e34T2 1 - 1.48e17T + 2.00e34T^{2}
47 1+9.51e16T+1.30e35T2 1 + 9.51e16T + 1.30e35T^{2}
53 1−1.03e18T+1.62e36T2 1 - 1.03e18T + 1.62e36T^{2}
59 1+1.03e18T+1.54e37T2 1 + 1.03e18T + 1.54e37T^{2}
61 1+8.22e18T+3.10e37T2 1 + 8.22e18T + 3.10e37T^{2}
67 1+1.74e19T+2.22e38T2 1 + 1.74e19T + 2.22e38T^{2}
71 1−2.67e19T+7.52e38T2 1 - 2.67e19T + 7.52e38T^{2}
73 1−5.60e19T+1.34e39T2 1 - 5.60e19T + 1.34e39T^{2}
79 1+4.33e19T+7.08e39T2 1 + 4.33e19T + 7.08e39T^{2}
83 1+2.59e20T+1.99e40T2 1 + 2.59e20T + 1.99e40T^{2}
89 1−5.09e20T+8.65e40T2 1 - 5.09e20T + 8.65e40T^{2}
97 1−6.58e20T+5.27e41T2 1 - 6.58e20T + 5.27e41T^{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

−15.80904391050943419199518934072, −14.16960218827182429796380241887, −12.44505953192757271287888041570, −11.61423277013174624547472955210, −10.16505969652227915066826270464, −7.63250874819259180382847280239, −6.01437335541878318939133969791, −4.89552429151551962596653638970, −3.10198697336722944316667572319, −0.794842427459680101765561788939, 0.794842427459680101765561788939, 3.10198697336722944316667572319, 4.89552429151551962596653638970, 6.01437335541878318939133969791, 7.63250874819259180382847280239, 10.16505969652227915066826270464, 11.61423277013174624547472955210, 12.44505953192757271287888041570, 14.16960218827182429796380241887, 15.80904391050943419199518934072

Graph of the ZZ-function along the critical line