Properties

Label 2-8001-1.1-c1-0-190
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  + 2.66·2-s + 5.10·4-s − 0.697·5-s − 7-s + 8.27·8-s − 1.86·10-s + 1.70·11-s − 0.0102·13-s − 2.66·14-s + 11.8·16-s + 3.99·17-s + 3.49·19-s − 3.56·20-s + 4.54·22-s + 0.546·23-s − 4.51·25-s − 0.0272·26-s − 5.10·28-s − 9.46·29-s + 7.67·31-s + 15.0·32-s + 10.6·34-s + 0.697·35-s + 7.99·37-s + 9.32·38-s − 5.77·40-s + 11.1·41-s + ⋯
L(s)  = 1  + 1.88·2-s + 2.55·4-s − 0.312·5-s − 0.377·7-s + 2.92·8-s − 0.588·10-s + 0.514·11-s − 0.00283·13-s − 0.712·14-s + 2.96·16-s + 0.968·17-s + 0.802·19-s − 0.796·20-s + 0.969·22-s + 0.114·23-s − 0.902·25-s − 0.00535·26-s − 0.964·28-s − 1.75·29-s + 1.37·31-s + 2.66·32-s + 1.82·34-s + 0.117·35-s + 1.31·37-s + 1.51·38-s − 0.913·40-s + 1.74·41-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

\(L(1)\) \(\approx\) \(7.558198287\)
\(L(\frac12)\) \(\approx\) \(7.558198287\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.66T + 2T^{2} \)
5 \( 1 + 0.697T + 5T^{2} \)
11 \( 1 - 1.70T + 11T^{2} \)
13 \( 1 + 0.0102T + 13T^{2} \)
17 \( 1 - 3.99T + 17T^{2} \)
19 \( 1 - 3.49T + 19T^{2} \)
23 \( 1 - 0.546T + 23T^{2} \)
29 \( 1 + 9.46T + 29T^{2} \)
31 \( 1 - 7.67T + 31T^{2} \)
37 \( 1 - 7.99T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 9.72T + 43T^{2} \)
47 \( 1 + 1.31T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + 6.94T + 59T^{2} \)
61 \( 1 + 5.17T + 61T^{2} \)
67 \( 1 - 15.8T + 67T^{2} \)
71 \( 1 - 1.12T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 - 6.13T + 89T^{2} \)
97 \( 1 + 6.18T + 97T^{2} \)
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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

−7.62044809876976566117667770164, −6.90883773760418782265945407711, −6.20015907411752279437669221336, −5.67474034131775441176033655734, −5.04082972003226559260352585603, −4.14249383009451493996172686340, −3.69992153247802390420194979741, −3.00937658244411407875966907500, −2.18136372095025102607535873303, −1.07487207832209839848710178834, 1.07487207832209839848710178834, 2.18136372095025102607535873303, 3.00937658244411407875966907500, 3.69992153247802390420194979741, 4.14249383009451493996172686340, 5.04082972003226559260352585603, 5.67474034131775441176033655734, 6.20015907411752279437669221336, 6.90883773760418782265945407711, 7.62044809876976566117667770164

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