Properties

Label 2-8001-1.1-c1-0-33
Degree $2$
Conductor $8001$
Sign $1$
Analytic cond. $63.8883$
Root an. cond. $7.99301$
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  − 1.87·2-s + 1.53·4-s − 2.15·5-s − 7-s + 0.882·8-s + 4.05·10-s + 1.05·11-s − 1.95·13-s + 1.87·14-s − 4.71·16-s + 3.66·17-s − 5.80·19-s − 3.30·20-s − 1.97·22-s − 0.881·23-s − 0.340·25-s + 3.67·26-s − 1.53·28-s + 7.03·29-s + 9.43·31-s + 7.10·32-s − 6.88·34-s + 2.15·35-s − 3.58·37-s + 10.9·38-s − 1.90·40-s + 8.10·41-s + ⋯
L(s)  = 1  − 1.32·2-s + 0.765·4-s − 0.965·5-s − 0.377·7-s + 0.312·8-s + 1.28·10-s + 0.317·11-s − 0.542·13-s + 0.502·14-s − 1.17·16-s + 0.888·17-s − 1.33·19-s − 0.738·20-s − 0.421·22-s − 0.183·23-s − 0.0681·25-s + 0.720·26-s − 0.289·28-s + 1.30·29-s + 1.69·31-s + 1.25·32-s − 1.18·34-s + 0.364·35-s − 0.588·37-s + 1.77·38-s − 0.301·40-s + 1.26·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
Sign: $1$
Analytic conductor: \(63.8883\)
Root analytic conductor: \(7.99301\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4454606289\)
\(L(\frac12)\) \(\approx\) \(0.4454606289\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
127 \( 1 + T \)
good2 \( 1 + 1.87T + 2T^{2} \)
5 \( 1 + 2.15T + 5T^{2} \)
11 \( 1 - 1.05T + 11T^{2} \)
13 \( 1 + 1.95T + 13T^{2} \)
17 \( 1 - 3.66T + 17T^{2} \)
19 \( 1 + 5.80T + 19T^{2} \)
23 \( 1 + 0.881T + 23T^{2} \)
29 \( 1 - 7.03T + 29T^{2} \)
31 \( 1 - 9.43T + 31T^{2} \)
37 \( 1 + 3.58T + 37T^{2} \)
41 \( 1 - 8.10T + 41T^{2} \)
43 \( 1 + 1.22T + 43T^{2} \)
47 \( 1 + 4.52T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 4.86T + 59T^{2} \)
61 \( 1 + 9.78T + 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 - 3.03T + 71T^{2} \)
73 \( 1 + 0.269T + 73T^{2} \)
79 \( 1 - 5.50T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−7.82489355181091327267055172166, −7.56869636175935366282947948389, −6.56992048239438673692109400535, −6.17673944005791403786663295827, −4.75793063133489394547261822611, −4.39887659752783349864617753469, −3.39345994254127788670792087784, −2.52733194493970615240874882934, −1.42705695921294308753669049316, −0.42916325068170793385182357402, 0.42916325068170793385182357402, 1.42705695921294308753669049316, 2.52733194493970615240874882934, 3.39345994254127788670792087784, 4.39887659752783349864617753469, 4.75793063133489394547261822611, 6.17673944005791403786663295827, 6.56992048239438673692109400535, 7.56869636175935366282947948389, 7.82489355181091327267055172166

Graph of the $Z$-function along the critical line