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Properties

Label 2-3971-1.1-c1-0-223
Degree 22
Conductor 39713971
Sign −1-1
Analytic cond. 31.708531.7085
Root an. cond. 5.631035.63103
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  + 1.23·2-s − 2.02·3-s − 0.482·4-s + 2.63·5-s − 2.49·6-s + 3.75·7-s − 3.05·8-s + 1.11·9-s + 3.24·10-s − 11-s + 0.978·12-s − 1.64·13-s + 4.62·14-s − 5.33·15-s − 2.80·16-s − 1.41·17-s + 1.36·18-s − 1.26·20-s − 7.62·21-s − 1.23·22-s − 2.42·23-s + 6.20·24-s + 1.92·25-s − 2.03·26-s + 3.82·27-s − 1.81·28-s + 7.21·29-s + ⋯
L(s)  = 1  + 0.871·2-s − 1.17·3-s − 0.241·4-s + 1.17·5-s − 1.01·6-s + 1.42·7-s − 1.08·8-s + 0.370·9-s + 1.02·10-s − 0.301·11-s + 0.282·12-s − 0.457·13-s + 1.23·14-s − 1.37·15-s − 0.700·16-s − 0.342·17-s + 0.322·18-s − 0.283·20-s − 1.66·21-s − 0.262·22-s − 0.505·23-s + 1.26·24-s + 0.384·25-s − 0.398·26-s + 0.736·27-s − 0.342·28-s + 1.33·29-s + ⋯

Functional equation

Λ(s)=(3971s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3971s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39713971    =    11⋅19211 \cdot 19^{2}
Sign: −1-1
Analytic conductor: 31.708531.7085
Root analytic conductor: 5.631035.63103
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3971, ( :1/2), −1)(2,\ 3971,\ (\ :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)
bad11 1+T 1 + T
19 1 1
good2 1−1.23T+2T2 1 - 1.23T + 2T^{2}
3 1+2.02T+3T2 1 + 2.02T + 3T^{2}
5 1−2.63T+5T2 1 - 2.63T + 5T^{2}
7 1−3.75T+7T2 1 - 3.75T + 7T^{2}
13 1+1.64T+13T2 1 + 1.64T + 13T^{2}
17 1+1.41T+17T2 1 + 1.41T + 17T^{2}
23 1+2.42T+23T2 1 + 2.42T + 23T^{2}
29 1−7.21T+29T2 1 - 7.21T + 29T^{2}
31 1+9.66T+31T2 1 + 9.66T + 31T^{2}
37 1+9.68T+37T2 1 + 9.68T + 37T^{2}
41 1+12.2T+41T2 1 + 12.2T + 41T^{2}
43 1+5.93T+43T2 1 + 5.93T + 43T^{2}
47 1−0.0866T+47T2 1 - 0.0866T + 47T^{2}
53 1+1.19T+53T2 1 + 1.19T + 53T^{2}
59 1+2.72T+59T2 1 + 2.72T + 59T^{2}
61 1−11.7T+61T2 1 - 11.7T + 61T^{2}
67 1−4.15T+67T2 1 - 4.15T + 67T^{2}
71 1−2.63T+71T2 1 - 2.63T + 71T^{2}
73 1−12.2T+73T2 1 - 12.2T + 73T^{2}
79 1−3.17T+79T2 1 - 3.17T + 79T^{2}
83 1+1.23T+83T2 1 + 1.23T + 83T^{2}
89 1+15.2T+89T2 1 + 15.2T + 89T^{2}
97 1+3.32T+97T2 1 + 3.32T + 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

−8.234609058282839573893145176082, −6.95273130391521232708365123988, −6.37745555182308308932353118086, −5.36077469066162409311095072548, −5.30667442258187139649416571842, −4.73556571190055223461658466814, −3.66454799691653072544041252701, −2.40228142487565428710539188251, −1.53159151173408042787073134414, 0, 1.53159151173408042787073134414, 2.40228142487565428710539188251, 3.66454799691653072544041252701, 4.73556571190055223461658466814, 5.30667442258187139649416571842, 5.36077469066162409311095072548, 6.37745555182308308932353118086, 6.95273130391521232708365123988, 8.234609058282839573893145176082

Graph of the ZZ-function along the critical line