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Properties

Label 2-8001-1.1-c1-0-216
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  − 0.446·2-s − 1.80·4-s + 2.33·5-s + 7-s + 1.69·8-s − 1.04·10-s − 4.53·11-s − 0.767·13-s − 0.446·14-s + 2.84·16-s − 3.43·17-s − 4.98·19-s − 4.20·20-s + 2.02·22-s + 4.14·23-s + 0.464·25-s + 0.342·26-s − 1.80·28-s + 9.37·29-s + 5.36·31-s − 4.66·32-s + 1.53·34-s + 2.33·35-s − 2.44·37-s + 2.22·38-s + 3.96·40-s + 1.04·41-s + ⋯
L(s)  = 1  − 0.315·2-s − 0.900·4-s + 1.04·5-s + 0.377·7-s + 0.599·8-s − 0.329·10-s − 1.36·11-s − 0.212·13-s − 0.119·14-s + 0.711·16-s − 0.832·17-s − 1.14·19-s − 0.941·20-s + 0.431·22-s + 0.864·23-s + 0.0929·25-s + 0.0671·26-s − 0.340·28-s + 1.74·29-s + 0.964·31-s − 0.824·32-s + 0.262·34-s + 0.395·35-s − 0.402·37-s + 0.361·38-s + 0.627·40-s + 0.163·41-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+0.446T+2T2 1 + 0.446T + 2T^{2}
5 1−2.33T+5T2 1 - 2.33T + 5T^{2}
11 1+4.53T+11T2 1 + 4.53T + 11T^{2}
13 1+0.767T+13T2 1 + 0.767T + 13T^{2}
17 1+3.43T+17T2 1 + 3.43T + 17T^{2}
19 1+4.98T+19T2 1 + 4.98T + 19T^{2}
23 1−4.14T+23T2 1 - 4.14T + 23T^{2}
29 1−9.37T+29T2 1 - 9.37T + 29T^{2}
31 1−5.36T+31T2 1 - 5.36T + 31T^{2}
37 1+2.44T+37T2 1 + 2.44T + 37T^{2}
41 1−1.04T+41T2 1 - 1.04T + 41T^{2}
43 1−2.90T+43T2 1 - 2.90T + 43T^{2}
47 1−1.19T+47T2 1 - 1.19T + 47T^{2}
53 1+10.5T+53T2 1 + 10.5T + 53T^{2}
59 1+1.68T+59T2 1 + 1.68T + 59T^{2}
61 1−4.42T+61T2 1 - 4.42T + 61T^{2}
67 1−11.8T+67T2 1 - 11.8T + 67T^{2}
71 1+5.39T+71T2 1 + 5.39T + 71T^{2}
73 1−5.28T+73T2 1 - 5.28T + 73T^{2}
79 1+5.18T+79T2 1 + 5.18T + 79T^{2}
83 1+6.33T+83T2 1 + 6.33T + 83T^{2}
89 1−1.35T+89T2 1 - 1.35T + 89T^{2}
97 1+8.94T+97T2 1 + 8.94T + 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

−7.68040346805330134450282221335, −6.74692088109717100893133845563, −6.09646288993958756066335749990, −5.19938423816290574469223720996, −4.84086135079742606336177749368, −4.13172646152765065805132861528, −2.83853220459818778705997419510, −2.25235849150977401716595798371, −1.18771676966119885437651168629, 0, 1.18771676966119885437651168629, 2.25235849150977401716595798371, 2.83853220459818778705997419510, 4.13172646152765065805132861528, 4.84086135079742606336177749368, 5.19938423816290574469223720996, 6.09646288993958756066335749990, 6.74692088109717100893133845563, 7.68040346805330134450282221335

Graph of the ZZ-function along the critical line