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Properties

Label 2-8002-1.1-c1-0-273
Degree $2$
Conductor $8002$
Sign $-1$
Analytic cond. $63.8962$
Root an. cond. $7.99351$
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-s + 4-s + 8-s − 3·9-s − 2·11-s + 4·13-s + 16-s − 3·18-s + 19-s − 2·22-s + 3·23-s − 5·25-s + 4·26-s − 4·29-s + 32-s − 3·36-s + 6·37-s + 38-s − 8·41-s − 12·43-s − 2·44-s + 3·46-s − 10·47-s − 7·49-s − 5·50-s + 4·52-s − 13·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 0.603·11-s + 1.10·13-s + 1/4·16-s − 0.707·18-s + 0.229·19-s − 0.426·22-s + 0.625·23-s − 25-s + 0.784·26-s − 0.742·29-s + 0.176·32-s − 1/2·36-s + 0.986·37-s + 0.162·38-s − 1.24·41-s − 1.82·43-s − 0.301·44-s + 0.442·46-s − 1.45·47-s − 49-s − 0.707·50-s + 0.554·52-s − 1.78·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8002\)    =    \(2 \cdot 4001\)
Sign: $-1$
Analytic conductor: \(63.8962\)
Root analytic conductor: \(7.99351\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8002,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
4001 \( 1+O(T) \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.55380803868944044771613529921, −6.36698766083733532203761326937, −6.28159301340715538102152901596, −5.17844599228044765088524077235, −4.96835862082833972226716777634, −3.62771983240159678724118102390, −3.36837093332359415208172087293, −2.37885053493444500995945992865, −1.45810296844493483995790663877, 0, 1.45810296844493483995790663877, 2.37885053493444500995945992865, 3.36837093332359415208172087293, 3.62771983240159678724118102390, 4.96835862082833972226716777634, 5.17844599228044765088524077235, 6.28159301340715538102152901596, 6.36698766083733532203761326937, 7.55380803868944044771613529921

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