Properties

Degree 2
Conductor $ 5 \cdot 1601 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 4-s + 5-s + 2·6-s + 2·7-s + 3·8-s + 9-s − 10-s − 6·11-s + 2·12-s − 2·13-s − 2·14-s − 2·15-s − 16-s + 6·17-s − 18-s − 4·19-s − 20-s − 4·21-s + 6·22-s + 2·23-s − 6·24-s + 25-s + 2·26-s + 4·27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s + 0.577·12-s − 0.554·13-s − 0.534·14-s − 0.516·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.872·21-s + 1.27·22-s + 0.417·23-s − 1.22·24-s + 1/5·25-s + 0.392·26-s + 0.769·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8005\)    =    \(5 \cdot 1601\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8005} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8005,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;1601\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;1601\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 - T \)
1601 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.64180625039070646647934243011, −6.96077607234823661197585781998, −5.81304454420980602866309778738, −5.36408462683488464840556406717, −4.99949483583419118726258580216, −4.20484602851035183485167770697, −2.91849881417646146763263909653, −1.96063183251360655991191717354, −0.909193834741079949871477253890, 0, 0.909193834741079949871477253890, 1.96063183251360655991191717354, 2.91849881417646146763263909653, 4.20484602851035183485167770697, 4.99949483583419118726258580216, 5.36408462683488464840556406717, 5.81304454420980602866309778738, 6.96077607234823661197585781998, 7.64180625039070646647934243011

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