Properties

Label 2-8001-1.1-c1-0-63
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  + 2.02·2-s + 2.10·4-s − 2.65·5-s − 7-s + 0.202·8-s − 5.36·10-s − 3.06·11-s + 0.621·13-s − 2.02·14-s − 3.78·16-s − 1.54·17-s + 7.42·19-s − 5.56·20-s − 6.19·22-s − 3.41·23-s + 2.02·25-s + 1.25·26-s − 2.10·28-s + 5.77·29-s − 8.60·31-s − 8.07·32-s − 3.13·34-s + 2.65·35-s + 8.63·37-s + 15.0·38-s − 0.537·40-s − 0.418·41-s + ⋯
L(s)  = 1  + 1.43·2-s + 1.05·4-s − 1.18·5-s − 0.377·7-s + 0.0717·8-s − 1.69·10-s − 0.923·11-s + 0.172·13-s − 0.541·14-s − 0.947·16-s − 0.375·17-s + 1.70·19-s − 1.24·20-s − 1.32·22-s − 0.712·23-s + 0.404·25-s + 0.246·26-s − 0.396·28-s + 1.07·29-s − 1.54·31-s − 1.42·32-s − 0.538·34-s + 0.447·35-s + 1.42·37-s + 2.43·38-s − 0.0850·40-s − 0.0653·41-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

\(L(1)\) \(\approx\) \(2.386081246\)
\(L(\frac12)\) \(\approx\) \(2.386081246\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.02T + 2T^{2} \)
5 \( 1 + 2.65T + 5T^{2} \)
11 \( 1 + 3.06T + 11T^{2} \)
13 \( 1 - 0.621T + 13T^{2} \)
17 \( 1 + 1.54T + 17T^{2} \)
19 \( 1 - 7.42T + 19T^{2} \)
23 \( 1 + 3.41T + 23T^{2} \)
29 \( 1 - 5.77T + 29T^{2} \)
31 \( 1 + 8.60T + 31T^{2} \)
37 \( 1 - 8.63T + 37T^{2} \)
41 \( 1 + 0.418T + 41T^{2} \)
43 \( 1 - 4.93T + 43T^{2} \)
47 \( 1 + 3.63T + 47T^{2} \)
53 \( 1 + 2.34T + 53T^{2} \)
59 \( 1 + 2.80T + 59T^{2} \)
61 \( 1 - 8.10T + 61T^{2} \)
67 \( 1 + 1.88T + 67T^{2} \)
71 \( 1 - 5.59T + 71T^{2} \)
73 \( 1 - 1.70T + 73T^{2} \)
79 \( 1 - 1.89T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−7.65275996942845717275049586112, −7.11234168000778288371001379973, −6.24531280628128396818916349844, −5.60158038801123685860315995788, −4.92730905629365655298355626770, −4.28389835655509139672917484974, −3.55368617517981808059342469743, −3.09560214529890195732363257693, −2.21024437610091664353550229741, −0.58526822642421430828757110663, 0.58526822642421430828757110663, 2.21024437610091664353550229741, 3.09560214529890195732363257693, 3.55368617517981808059342469743, 4.28389835655509139672917484974, 4.92730905629365655298355626770, 5.60158038801123685860315995788, 6.24531280628128396818916349844, 7.11234168000778288371001379973, 7.65275996942845717275049586112

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