Properties

Label 2-4001-1.1-c1-0-10
Degree $2$
Conductor $4001$
Sign $1$
Analytic cond. $31.9481$
Root an. cond. $5.65226$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(4001\)
Sign: $1$
Analytic conductor: \(31.9481\)
Root analytic conductor: \(5.65226\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02088209983\)
\(L(\frac12)\) \(\approx\) \(0.02088209983\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1+O(T) \)
good2 \( 1 + 1.22T + 2T^{2} \)
3 \( 1 + 3.26T + 3T^{2} \)
5 \( 1 + 3.92T + 5T^{2} \)
7 \( 1 + 1.66T + 7T^{2} \)
11 \( 1 + 2.19T + 11T^{2} \)
13 \( 1 + 4.23T + 13T^{2} \)
17 \( 1 + 2.75T + 17T^{2} \)
19 \( 1 - 6.81T + 19T^{2} \)
23 \( 1 - 0.282T + 23T^{2} \)
29 \( 1 + 3.43T + 29T^{2} \)
31 \( 1 - 9.87T + 31T^{2} \)
37 \( 1 - 3.32T + 37T^{2} \)
41 \( 1 - 4.80T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + 8.15T + 47T^{2} \)
53 \( 1 + 0.816T + 53T^{2} \)
59 \( 1 - 3.51T + 59T^{2} \)
61 \( 1 + 9.36T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 + 7.27T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 7.05T + 83T^{2} \)
89 \( 1 + 7.20T + 89T^{2} \)
97 \( 1 - 7.79T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line