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Properties

Label 2-4001-1.1-c1-0-10
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  − 1.22·2-s − 3.26·3-s − 0.507·4-s − 3.92·5-s + 3.99·6-s − 1.66·7-s + 3.06·8-s + 7.67·9-s + 4.79·10-s − 2.19·11-s + 1.65·12-s − 4.23·13-s + 2.03·14-s + 12.8·15-s − 2.72·16-s − 2.75·17-s − 9.38·18-s + 6.81·19-s + 1.99·20-s + 5.45·21-s + 2.67·22-s + 0.282·23-s − 10.0·24-s + 10.4·25-s + 5.17·26-s − 15.2·27-s + 0.845·28-s + ⋯
L(s)  = 1  − 0.863·2-s − 1.88·3-s − 0.253·4-s − 1.75·5-s + 1.63·6-s − 0.630·7-s + 1.08·8-s + 2.55·9-s + 1.51·10-s − 0.660·11-s + 0.478·12-s − 1.17·13-s + 0.544·14-s + 3.31·15-s − 0.682·16-s − 0.669·17-s − 2.21·18-s + 1.56·19-s + 0.445·20-s + 1.18·21-s + 0.570·22-s + 0.0588·23-s − 2.04·24-s + 2.08·25-s + 1.01·26-s − 2.94·27-s + 0.159·28-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 0.020882099830.02088209983
L(12)L(\frac12) ≈\approx 0.020882099830.02088209983
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+1.22T+2T2 1 + 1.22T + 2T^{2}
3 1+3.26T+3T2 1 + 3.26T + 3T^{2}
5 1+3.92T+5T2 1 + 3.92T + 5T^{2}
7 1+1.66T+7T2 1 + 1.66T + 7T^{2}
11 1+2.19T+11T2 1 + 2.19T + 11T^{2}
13 1+4.23T+13T2 1 + 4.23T + 13T^{2}
17 1+2.75T+17T2 1 + 2.75T + 17T^{2}
19 1−6.81T+19T2 1 - 6.81T + 19T^{2}
23 1−0.282T+23T2 1 - 0.282T + 23T^{2}
29 1+3.43T+29T2 1 + 3.43T + 29T^{2}
31 1−9.87T+31T2 1 - 9.87T + 31T^{2}
37 1−3.32T+37T2 1 - 3.32T + 37T^{2}
41 1−4.80T+41T2 1 - 4.80T + 41T^{2}
43 1+11.7T+43T2 1 + 11.7T + 43T^{2}
47 1+8.15T+47T2 1 + 8.15T + 47T^{2}
53 1+0.816T+53T2 1 + 0.816T + 53T^{2}
59 1−3.51T+59T2 1 - 3.51T + 59T^{2}
61 1+9.36T+61T2 1 + 9.36T + 61T^{2}
67 1+11.8T+67T2 1 + 11.8T + 67T^{2}
71 1+7.27T+71T2 1 + 7.27T + 71T^{2}
73 1−11.5T+73T2 1 - 11.5T + 73T^{2}
79 1+10.5T+79T2 1 + 10.5T + 79T^{2}
83 1+7.05T+83T2 1 + 7.05T + 83T^{2}
89 1+7.20T+89T2 1 + 7.20T + 89T^{2}
97 1−7.79T+97T2 1 - 7.79T + 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

−8.214150284917944401006904090319, −7.62442275416012400805683043174, −7.15377489861792966154194359589, −6.46231846577106322552763331802, −5.27564382438385023740430952274, −4.74839881938934649240885321177, −4.21994267292475963353349162797, −3.06332789165973178934312861795, −1.19670373244097194793651251215, −0.12250324289815012062015376768, 0.12250324289815012062015376768, 1.19670373244097194793651251215, 3.06332789165973178934312861795, 4.21994267292475963353349162797, 4.74839881938934649240885321177, 5.27564382438385023740430952274, 6.46231846577106322552763331802, 7.15377489861792966154194359589, 7.62442275416012400805683043174, 8.214150284917944401006904090319

Graph of the ZZ-function along the critical line