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Properties

Label 2-8001-1.1-c1-0-285
Degree 22
Conductor 80018001
Sign −1-1
Analytic cond. 63.888363.8883
Root an. cond. 7.993017.99301
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  + 2.37·2-s + 3.65·4-s − 2.39·5-s + 7-s + 3.92·8-s − 5.68·10-s − 1.77·11-s − 3.89·13-s + 2.37·14-s + 2.03·16-s + 5.42·17-s + 2.10·19-s − 8.72·20-s − 4.22·22-s − 4.43·23-s + 0.713·25-s − 9.25·26-s + 3.65·28-s + 5.14·29-s − 3.27·31-s − 3.02·32-s + 12.8·34-s − 2.39·35-s − 6.54·37-s + 5.00·38-s − 9.38·40-s + 5.69·41-s + ⋯
L(s)  = 1  + 1.68·2-s + 1.82·4-s − 1.06·5-s + 0.377·7-s + 1.38·8-s − 1.79·10-s − 0.536·11-s − 1.07·13-s + 0.635·14-s + 0.507·16-s + 1.31·17-s + 0.483·19-s − 1.95·20-s − 0.901·22-s − 0.924·23-s + 0.142·25-s − 1.81·26-s + 0.690·28-s + 0.955·29-s − 0.587·31-s − 0.534·32-s + 2.21·34-s − 0.404·35-s − 1.07·37-s + 0.812·38-s − 1.48·40-s + 0.888·41-s + ⋯

Functional equation

Λ(s)=(8001s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8001s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 80018001    =    32⋅7⋅1273^{2} \cdot 7 \cdot 127
Sign: −1-1
Analytic conductor: 63.888363.8883
Root analytic conductor: 7.993017.99301
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8001, ( :1/2), −1)(2,\ 8001,\ (\ :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)
bad3 1 1
7 1−T 1 - T
127 1−T 1 - T
good2 1−2.37T+2T2 1 - 2.37T + 2T^{2}
5 1+2.39T+5T2 1 + 2.39T + 5T^{2}
11 1+1.77T+11T2 1 + 1.77T + 11T^{2}
13 1+3.89T+13T2 1 + 3.89T + 13T^{2}
17 1−5.42T+17T2 1 - 5.42T + 17T^{2}
19 1−2.10T+19T2 1 - 2.10T + 19T^{2}
23 1+4.43T+23T2 1 + 4.43T + 23T^{2}
29 1−5.14T+29T2 1 - 5.14T + 29T^{2}
31 1+3.27T+31T2 1 + 3.27T + 31T^{2}
37 1+6.54T+37T2 1 + 6.54T + 37T^{2}
41 1−5.69T+41T2 1 - 5.69T + 41T^{2}
43 1+1.32T+43T2 1 + 1.32T + 43T^{2}
47 1+1.09T+47T2 1 + 1.09T + 47T^{2}
53 1+9.83T+53T2 1 + 9.83T + 53T^{2}
59 1+5.41T+59T2 1 + 5.41T + 59T^{2}
61 1−7.48T+61T2 1 - 7.48T + 61T^{2}
67 1−9.52T+67T2 1 - 9.52T + 67T^{2}
71 1+9.03T+71T2 1 + 9.03T + 71T^{2}
73 1+15.3T+73T2 1 + 15.3T + 73T^{2}
79 1+10.0T+79T2 1 + 10.0T + 79T^{2}
83 1+13.7T+83T2 1 + 13.7T + 83T^{2}
89 1+11.1T+89T2 1 + 11.1T + 89T^{2}
97 1+15.9T+97T2 1 + 15.9T + 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

−7.44484196516352913026121320953, −6.75603824266903032722437347735, −5.72144580215065492584013618041, −5.34555430833845514831595042898, −4.56359301855515302550748463402, −4.06808181788419093542321504912, −3.23147485052135509123937839493, −2.71316038960204569578935121911, −1.60021999321748527847793507163, 0, 1.60021999321748527847793507163, 2.71316038960204569578935121911, 3.23147485052135509123937839493, 4.06808181788419093542321504912, 4.56359301855515302550748463402, 5.34555430833845514831595042898, 5.72144580215065492584013618041, 6.75603824266903032722437347735, 7.44484196516352913026121320953

Graph of the ZZ-function along the critical line