Properties

Degree $2$
Conductor $8001$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.44·2-s + 3.99·4-s + 2.44·5-s + 7-s + 4.89·8-s + 5.99·10-s + 6.89·13-s + 2.44·14-s + 3.99·16-s + 0.550·17-s − 0.449·19-s + 9.79·20-s − 6·23-s + 0.999·25-s + 16.8·26-s + 3.99·28-s − 1.89·29-s + 8·31-s + 1.34·34-s + 2.44·35-s − 7·37-s − 1.10·38-s + 11.9·40-s − 0.550·41-s + 2·43-s − 14.6·46-s + 12·47-s + ⋯
L(s)  = 1  + 1.73·2-s + 1.99·4-s + 1.09·5-s + 0.377·7-s + 1.73·8-s + 1.89·10-s + 1.91·13-s + 0.654·14-s + 0.999·16-s + 0.133·17-s − 0.103·19-s + 2.19·20-s − 1.25·23-s + 0.199·25-s + 3.31·26-s + 0.755·28-s − 0.352·29-s + 1.43·31-s + 0.231·34-s + 0.414·35-s − 1.15·37-s − 0.178·38-s + 1.89·40-s − 0.0859·41-s + 0.304·43-s − 2.16·46-s + 1.75·47-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: $1$
Motivic weight: \(1\)
Character: $\chi_{8001} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.896678798\)
\(L(\frac12)\) \(\approx\) \(8.896678798\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.44T + 2T^{2} \)
5 \( 1 - 2.44T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.89T + 13T^{2} \)
17 \( 1 - 0.550T + 17T^{2} \)
19 \( 1 + 0.449T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 1.89T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + 0.550T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 7.89T + 53T^{2} \)
59 \( 1 + 2.44T + 59T^{2} \)
61 \( 1 + 0.449T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 + 8.44T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 5.89T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 9.79T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−7.59142499423268438437091792175, −6.73242015990030196977210017948, −6.03547288909840762799806781183, −5.83777342285728021511503054030, −5.13824239300944879145112804146, −4.16772098887322074841561776458, −3.78876775369904540773722427496, −2.80222178944816551373719985819, −2.03780219137573842424714449543, −1.28266928430766931766973504102, 1.28266928430766931766973504102, 2.03780219137573842424714449543, 2.80222178944816551373719985819, 3.78876775369904540773722427496, 4.16772098887322074841561776458, 5.13824239300944879145112804146, 5.83777342285728021511503054030, 6.03547288909840762799806781183, 6.73242015990030196977210017948, 7.59142499423268438437091792175

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