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Properties

Label 2-8001-1.1-c1-0-283
Degree $2$
Conductor $8001$
Sign $-1$
Analytic cond. $63.8883$
Root an. cond. $7.99301$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 7-s + 2·13-s − 2·14-s − 4·16-s − 17-s − 2·19-s − 5·25-s + 4·26-s − 2·28-s + 9·29-s − 8·32-s − 2·34-s − 37-s − 4·38-s + 3·41-s − 6·43-s − 8·47-s + 49-s − 10·50-s + 4·52-s − 11·53-s + 18·58-s + 12·59-s − 10·61-s − 8·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.377·7-s + 0.554·13-s − 0.534·14-s − 16-s − 0.242·17-s − 0.458·19-s − 25-s + 0.784·26-s − 0.377·28-s + 1.67·29-s − 1.41·32-s − 0.342·34-s − 0.164·37-s − 0.648·38-s + 0.468·41-s − 0.914·43-s − 1.16·47-s + 1/7·49-s − 1.41·50-s + 0.554·52-s − 1.51·53-s + 2.36·58-s + 1.56·59-s − 1.28·61-s − 64-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8001,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−7.18329511887227323587743097188, −6.45572513992513184753117667402, −6.12002890837399600947111827811, −5.31164241113513451023753715761, −4.58491860612885164291031051030, −4.01537396309127377486690529513, −3.24074703911987210459280825914, −2.61250521893054700120726497817, −1.54963313429784272002828311844, 0, 1.54963313429784272002828311844, 2.61250521893054700120726497817, 3.24074703911987210459280825914, 4.01537396309127377486690529513, 4.58491860612885164291031051030, 5.31164241113513451023753715761, 6.12002890837399600947111827811, 6.45572513992513184753117667402, 7.18329511887227323587743097188

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