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Properties

Label 2-3971-1.1-c1-0-170
Degree 22
Conductor 39713971
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.09·2-s + 2.89·3-s + 2.38·4-s − 3.61·5-s + 6.05·6-s + 0.264·7-s + 0.802·8-s + 5.36·9-s − 7.55·10-s + 11-s + 6.89·12-s + 5.66·13-s + 0.554·14-s − 10.4·15-s − 3.08·16-s + 0.577·17-s + 11.2·18-s − 8.60·20-s + 0.766·21-s + 2.09·22-s + 2.58·23-s + 2.32·24-s + 8.03·25-s + 11.8·26-s + 6.84·27-s + 0.631·28-s + 7.43·29-s + ⋯
L(s)  = 1  + 1.48·2-s + 1.67·3-s + 1.19·4-s − 1.61·5-s + 2.47·6-s + 0.100·7-s + 0.283·8-s + 1.78·9-s − 2.39·10-s + 0.301·11-s + 1.98·12-s + 1.57·13-s + 0.148·14-s − 2.69·15-s − 0.771·16-s + 0.140·17-s + 2.64·18-s − 1.92·20-s + 0.167·21-s + 0.446·22-s + 0.539·23-s + 0.473·24-s + 1.60·25-s + 2.32·26-s + 1.31·27-s + 0.119·28-s + 1.38·29-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: 11
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: 00
Selberg data: (2, 3971, ( :1/2), 1)(2,\ 3971,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.6543718826.654371882
L(12)L(\frac12) ≈\approx 6.6543718826.654371882
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 1−2.09T+2T2 1 - 2.09T + 2T^{2}
3 1−2.89T+3T2 1 - 2.89T + 3T^{2}
5 1+3.61T+5T2 1 + 3.61T + 5T^{2}
7 1−0.264T+7T2 1 - 0.264T + 7T^{2}
13 1−5.66T+13T2 1 - 5.66T + 13T^{2}
17 1−0.577T+17T2 1 - 0.577T + 17T^{2}
23 1−2.58T+23T2 1 - 2.58T + 23T^{2}
29 1−7.43T+29T2 1 - 7.43T + 29T^{2}
31 1−9.21T+31T2 1 - 9.21T + 31T^{2}
37 1+2.09T+37T2 1 + 2.09T + 37T^{2}
41 1−0.639T+41T2 1 - 0.639T + 41T^{2}
43 1−5.69T+43T2 1 - 5.69T + 43T^{2}
47 1+4.15T+47T2 1 + 4.15T + 47T^{2}
53 1+5.18T+53T2 1 + 5.18T + 53T^{2}
59 1−4.63T+59T2 1 - 4.63T + 59T^{2}
61 1+4.89T+61T2 1 + 4.89T + 61T^{2}
67 1−8.70T+67T2 1 - 8.70T + 67T^{2}
71 1−3.21T+71T2 1 - 3.21T + 71T^{2}
73 1−14.8T+73T2 1 - 14.8T + 73T^{2}
79 1−2.18T+79T2 1 - 2.18T + 79T^{2}
83 1−2.42T+83T2 1 - 2.42T + 83T^{2}
89 1+18.0T+89T2 1 + 18.0T + 89T^{2}
97 1+15.4T+97T2 1 + 15.4T + 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

−8.243261883665974773754177397806, −7.956910211790867722598164137454, −6.85074093403001164802849937724, −6.42695095606757021911975160421, −5.07617787408961151208254331274, −4.24941601220466938477238866653, −3.87141055446800545503681276421, −3.21101500119157364240657357836, −2.67308558508209002441985888390, −1.20508341376257433977586716432, 1.20508341376257433977586716432, 2.67308558508209002441985888390, 3.21101500119157364240657357836, 3.87141055446800545503681276421, 4.24941601220466938477238866653, 5.07617787408961151208254331274, 6.42695095606757021911975160421, 6.85074093403001164802849937724, 7.956910211790867722598164137454, 8.243261883665974773754177397806

Graph of the ZZ-function along the critical line