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Properties

Label 2-4001-1.1-c1-0-142
Degree 22
Conductor 40014001
Sign 11
Analytic cond. 31.948131.9481
Root an. cond. 5.652265.65226
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  − 2.76·2-s − 0.782·3-s + 5.63·4-s + 4.06·5-s + 2.16·6-s + 1.66·7-s − 10.0·8-s − 2.38·9-s − 11.2·10-s + 5.36·11-s − 4.41·12-s + 4.72·13-s − 4.59·14-s − 3.17·15-s + 16.5·16-s − 0.338·17-s + 6.59·18-s − 2.39·19-s + 22.8·20-s − 1.30·21-s − 14.8·22-s + 1.52·23-s + 7.86·24-s + 11.4·25-s − 13.0·26-s + 4.21·27-s + 9.37·28-s + ⋯
L(s)  = 1  − 1.95·2-s − 0.451·3-s + 2.81·4-s + 1.81·5-s + 0.883·6-s + 0.628·7-s − 3.55·8-s − 0.795·9-s − 3.54·10-s + 1.61·11-s − 1.27·12-s + 1.31·13-s − 1.22·14-s − 0.820·15-s + 4.12·16-s − 0.0821·17-s + 1.55·18-s − 0.548·19-s + 5.11·20-s − 0.284·21-s − 3.16·22-s + 0.317·23-s + 1.60·24-s + 2.29·25-s − 2.56·26-s + 0.811·27-s + 1.77·28-s + ⋯

Functional equation

Λ(s)=(4001s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4001s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40014001
Sign: 11
Analytic conductor: 31.948131.9481
Root analytic conductor: 5.652265.65226
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4001, ( :1/2), 1)(2,\ 4001,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2433515771.243351577
L(12)L(\frac12) ≈\approx 1.2433515771.243351577
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)
bad4001 1+O(T) 1+O(T)
good2 1+2.76T+2T2 1 + 2.76T + 2T^{2}
3 1+0.782T+3T2 1 + 0.782T + 3T^{2}
5 1−4.06T+5T2 1 - 4.06T + 5T^{2}
7 1−1.66T+7T2 1 - 1.66T + 7T^{2}
11 1−5.36T+11T2 1 - 5.36T + 11T^{2}
13 1−4.72T+13T2 1 - 4.72T + 13T^{2}
17 1+0.338T+17T2 1 + 0.338T + 17T^{2}
19 1+2.39T+19T2 1 + 2.39T + 19T^{2}
23 1−1.52T+23T2 1 - 1.52T + 23T^{2}
29 1+2.87T+29T2 1 + 2.87T + 29T^{2}
31 1−7.10T+31T2 1 - 7.10T + 31T^{2}
37 1−7.03T+37T2 1 - 7.03T + 37T^{2}
41 1+7.06T+41T2 1 + 7.06T + 41T^{2}
43 1+10.7T+43T2 1 + 10.7T + 43T^{2}
47 1+7.82T+47T2 1 + 7.82T + 47T^{2}
53 1−13.0T+53T2 1 - 13.0T + 53T^{2}
59 1−0.352T+59T2 1 - 0.352T + 59T^{2}
61 1+2.23T+61T2 1 + 2.23T + 61T^{2}
67 1−8.18T+67T2 1 - 8.18T + 67T^{2}
71 1−9.16T+71T2 1 - 9.16T + 71T^{2}
73 1+9.95T+73T2 1 + 9.95T + 73T^{2}
79 1−14.1T+79T2 1 - 14.1T + 79T^{2}
83 1+2.57T+83T2 1 + 2.57T + 83T^{2}
89 1−15.4T+89T2 1 - 15.4T + 89T^{2}
97 1−2.79T+97T2 1 - 2.79T + 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

−8.611451523562219188660092945902, −8.162256662934724363611508458382, −6.74157705976395324965641711153, −6.44249763571049670715921406664, −6.01558017309345214091914271609, −5.09856258016982993562850627552, −3.42622350682267399124785557953, −2.33441621603393359648992819788, −1.56961674514797472934814529366, −0.975566362556027713750780891742, 0.975566362556027713750780891742, 1.56961674514797472934814529366, 2.33441621603393359648992819788, 3.42622350682267399124785557953, 5.09856258016982993562850627552, 6.01558017309345214091914271609, 6.44249763571049670715921406664, 6.74157705976395324965641711153, 8.162256662934724363611508458382, 8.611451523562219188660092945902

Graph of the ZZ-function along the critical line