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Properties

Label 2-91e2-1.1-c1-0-100
Degree 22
Conductor 82818281
Sign 11
Analytic cond. 66.124166.1241
Root an. cond. 8.131678.13167
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  − 1.09·2-s − 1.39·3-s − 0.807·4-s − 1.62·5-s + 1.52·6-s + 3.06·8-s − 1.05·9-s + 1.77·10-s + 1.29·11-s + 1.12·12-s + 2.26·15-s − 1.73·16-s + 5.41·17-s + 1.14·18-s − 1.51·19-s + 1.31·20-s − 1.41·22-s − 2.64·23-s − 4.27·24-s − 2.35·25-s + 5.65·27-s + 5.81·29-s − 2.47·30-s + 7.28·31-s − 4.23·32-s − 1.80·33-s − 5.90·34-s + ⋯
L(s)  = 1  − 0.772·2-s − 0.805·3-s − 0.403·4-s − 0.727·5-s + 0.622·6-s + 1.08·8-s − 0.350·9-s + 0.561·10-s + 0.390·11-s + 0.325·12-s + 0.586·15-s − 0.433·16-s + 1.31·17-s + 0.270·18-s − 0.346·19-s + 0.293·20-s − 0.301·22-s − 0.550·23-s − 0.873·24-s − 0.470·25-s + 1.08·27-s + 1.08·29-s − 0.452·30-s + 1.30·31-s − 0.749·32-s − 0.314·33-s − 1.01·34-s + ⋯

Functional equation

Λ(s)=(8281s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8281s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 82818281    =    72⋅1327^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 66.124166.1241
Root analytic conductor: 8.131678.13167
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8281, ( :1/2), 1)(2,\ 8281,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.60420684260.6042068426
L(12)L(\frac12) ≈\approx 0.60420684260.6042068426
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)
bad7 1 1
13 1 1
good2 1+1.09T+2T2 1 + 1.09T + 2T^{2}
3 1+1.39T+3T2 1 + 1.39T + 3T^{2}
5 1+1.62T+5T2 1 + 1.62T + 5T^{2}
11 1−1.29T+11T2 1 - 1.29T + 11T^{2}
17 1−5.41T+17T2 1 - 5.41T + 17T^{2}
19 1+1.51T+19T2 1 + 1.51T + 19T^{2}
23 1+2.64T+23T2 1 + 2.64T + 23T^{2}
29 1−5.81T+29T2 1 - 5.81T + 29T^{2}
31 1−7.28T+31T2 1 - 7.28T + 31T^{2}
37 1−6.95T+37T2 1 - 6.95T + 37T^{2}
41 1+8.09T+41T2 1 + 8.09T + 41T^{2}
43 1−11.1T+43T2 1 - 11.1T + 43T^{2}
47 1−7.17T+47T2 1 - 7.17T + 47T^{2}
53 1−4.66T+53T2 1 - 4.66T + 53T^{2}
59 1−0.773T+59T2 1 - 0.773T + 59T^{2}
61 1+8.74T+61T2 1 + 8.74T + 61T^{2}
67 1+6.37T+67T2 1 + 6.37T + 67T^{2}
71 1+11.9T+71T2 1 + 11.9T + 71T^{2}
73 1−7.21T+73T2 1 - 7.21T + 73T^{2}
79 1+11.7T+79T2 1 + 11.7T + 79T^{2}
83 1−8.42T+83T2 1 - 8.42T + 83T^{2}
89 1+1.66T+89T2 1 + 1.66T + 89T^{2}
97 1−12.4T+97T2 1 - 12.4T + 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

−7.72792911789640590073492791498, −7.47727290614848447334733171690, −6.30739567541096766403262053037, −5.89762849235619401581026237221, −4.93689194756712455859006501874, −4.37935660453765164003047917938, −3.63208102599414868160534190036, −2.60582284440693825623472518388, −1.22195767521668794103994724812, −0.53227070040051855052438261630, 0.53227070040051855052438261630, 1.22195767521668794103994724812, 2.60582284440693825623472518388, 3.63208102599414868160534190036, 4.37935660453765164003047917938, 4.93689194756712455859006501874, 5.89762849235619401581026237221, 6.30739567541096766403262053037, 7.47727290614848447334733171690, 7.72792911789640590073492791498

Graph of the ZZ-function along the critical line