Properties

Label 2-8001-1.1-c1-0-207
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  + 2.72·2-s + 5.44·4-s − 0.297·5-s − 7-s + 9.39·8-s − 0.811·10-s − 1.00·11-s + 6.53·13-s − 2.72·14-s + 14.7·16-s − 0.490·17-s + 1.20·19-s − 1.61·20-s − 2.74·22-s + 5.71·23-s − 4.91·25-s + 17.8·26-s − 5.44·28-s − 2.54·29-s + 1.43·31-s + 21.4·32-s − 1.33·34-s + 0.297·35-s − 4.61·37-s + 3.27·38-s − 2.79·40-s + 7.46·41-s + ⋯
L(s)  = 1  + 1.92·2-s + 2.72·4-s − 0.133·5-s − 0.377·7-s + 3.32·8-s − 0.256·10-s − 0.303·11-s + 1.81·13-s − 0.729·14-s + 3.68·16-s − 0.118·17-s + 0.275·19-s − 0.362·20-s − 0.585·22-s + 1.19·23-s − 0.982·25-s + 3.49·26-s − 1.02·28-s − 0.472·29-s + 0.256·31-s + 3.78·32-s − 0.229·34-s + 0.0502·35-s − 0.759·37-s + 0.531·38-s − 0.441·40-s + 1.16·41-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

\(L(1)\) \(\approx\) \(8.346891071\)
\(L(\frac12)\) \(\approx\) \(8.346891071\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.72T + 2T^{2} \)
5 \( 1 + 0.297T + 5T^{2} \)
11 \( 1 + 1.00T + 11T^{2} \)
13 \( 1 - 6.53T + 13T^{2} \)
17 \( 1 + 0.490T + 17T^{2} \)
19 \( 1 - 1.20T + 19T^{2} \)
23 \( 1 - 5.71T + 23T^{2} \)
29 \( 1 + 2.54T + 29T^{2} \)
31 \( 1 - 1.43T + 31T^{2} \)
37 \( 1 + 4.61T + 37T^{2} \)
41 \( 1 - 7.46T + 41T^{2} \)
43 \( 1 - 2.51T + 43T^{2} \)
47 \( 1 + 5.66T + 47T^{2} \)
53 \( 1 + 1.09T + 53T^{2} \)
59 \( 1 - 2.52T + 59T^{2} \)
61 \( 1 - 6.71T + 61T^{2} \)
67 \( 1 - 4.35T + 67T^{2} \)
71 \( 1 + 9.33T + 71T^{2} \)
73 \( 1 + 4.25T + 73T^{2} \)
79 \( 1 - 5.38T + 79T^{2} \)
83 \( 1 - 2.93T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 - 2.21T + 97T^{2} \)
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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

−7.49795502979784506341052671261, −6.87712286332996599910727151754, −6.16336223627607650655120514085, −5.74559889973396710914038747772, −5.02420725619504270913853969710, −4.20179585621242335571101184499, −3.59521045613249633321821958305, −3.07916535785199760188350208050, −2.13188036460854583979269878181, −1.14790023615198122733707773993, 1.14790023615198122733707773993, 2.13188036460854583979269878181, 3.07916535785199760188350208050, 3.59521045613249633321821958305, 4.20179585621242335571101184499, 5.02420725619504270913853969710, 5.74559889973396710914038747772, 6.16336223627607650655120514085, 6.87712286332996599910727151754, 7.49795502979784506341052671261

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