Properties

Label 2-8001-1.1-c1-0-22
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  − 1.45·2-s + 0.107·4-s − 0.584·5-s − 7-s + 2.74·8-s + 0.848·10-s + 5.41·11-s − 3.18·13-s + 1.45·14-s − 4.20·16-s − 5.06·17-s − 5.11·19-s − 0.0631·20-s − 7.86·22-s − 3.75·23-s − 4.65·25-s + 4.62·26-s − 0.107·28-s − 8.71·29-s + 3.51·31-s + 0.610·32-s + 7.34·34-s + 0.584·35-s − 2.23·37-s + 7.42·38-s − 1.60·40-s + 6.65·41-s + ⋯
L(s)  = 1  − 1.02·2-s + 0.0539·4-s − 0.261·5-s − 0.377·7-s + 0.971·8-s + 0.268·10-s + 1.63·11-s − 0.882·13-s + 0.388·14-s − 1.05·16-s − 1.22·17-s − 1.17·19-s − 0.0141·20-s − 1.67·22-s − 0.783·23-s − 0.931·25-s + 0.906·26-s − 0.0204·28-s − 1.61·29-s + 0.631·31-s + 0.107·32-s + 1.26·34-s + 0.0988·35-s − 0.367·37-s + 1.20·38-s − 0.253·40-s + 1.03·41-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

\(L(1)\) \(\approx\) \(0.4283854454\)
\(L(\frac12)\) \(\approx\) \(0.4283854454\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.45T + 2T^{2} \)
5 \( 1 + 0.584T + 5T^{2} \)
11 \( 1 - 5.41T + 11T^{2} \)
13 \( 1 + 3.18T + 13T^{2} \)
17 \( 1 + 5.06T + 17T^{2} \)
19 \( 1 + 5.11T + 19T^{2} \)
23 \( 1 + 3.75T + 23T^{2} \)
29 \( 1 + 8.71T + 29T^{2} \)
31 \( 1 - 3.51T + 31T^{2} \)
37 \( 1 + 2.23T + 37T^{2} \)
41 \( 1 - 6.65T + 41T^{2} \)
43 \( 1 + 4.51T + 43T^{2} \)
47 \( 1 + 9.14T + 47T^{2} \)
53 \( 1 + 2.10T + 53T^{2} \)
59 \( 1 - 5.83T + 59T^{2} \)
61 \( 1 - 14.0T + 61T^{2} \)
67 \( 1 + 3.86T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 3.62T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 - 6.96T + 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−8.038740071560444533207503795267, −7.16581401328221066968908330942, −6.69445838126875083877781237701, −6.01121144918516224197956052257, −4.89516580333750244021210278257, −4.11819786209088161364264346221, −3.77386578814433512821724301002, −2.27841065739508341763695455909, −1.69464559630353517062665678638, −0.37611993757242477302058544491, 0.37611993757242477302058544491, 1.69464559630353517062665678638, 2.27841065739508341763695455909, 3.77386578814433512821724301002, 4.11819786209088161364264346221, 4.89516580333750244021210278257, 6.01121144918516224197956052257, 6.69445838126875083877781237701, 7.16581401328221066968908330942, 8.038740071560444533207503795267

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