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Properties

Label 2-8002-1.1-c1-0-265
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 − 3-s + 4-s + 5-s − 6-s + 8-s − 2·9-s + 10-s − 12-s + 2·13-s − 15-s + 16-s + 4·17-s − 2·18-s − 4·19-s + 20-s + 23-s − 24-s − 4·25-s + 2·26-s + 5·27-s − 6·29-s − 30-s − 6·31-s + 32-s + 4·34-s − 2·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.471·18-s − 0.917·19-s + 0.223·20-s + 0.208·23-s − 0.204·24-s − 4/5·25-s + 0.392·26-s + 0.962·27-s − 1.11·29-s − 0.182·30-s − 1.07·31-s + 0.176·32-s + 0.685·34-s − 1/3·36-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 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−7.40412466481654531769612822275, −6.46946237492347963599767431716, −5.98951719859661503470343659228, −5.49143635302604460959945084676, −4.85690147301130752418385293024, −3.85624988113149508067351157939, −3.27667742137896483243809010935, −2.26044893917032892950267994783, −1.42397877824057140087387322920, 0, 1.42397877824057140087387322920, 2.26044893917032892950267994783, 3.27667742137896483243809010935, 3.85624988113149508067351157939, 4.85690147301130752418385293024, 5.49143635302604460959945084676, 5.98951719859661503470343659228, 6.46946237492347963599767431716, 7.40412466481654531769612822275

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