Properties

Degree 2
Conductor $ 5 \cdot 401 $
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 − 4-s + 5-s + 3·8-s − 3·9-s − 10-s + 4·11-s − 2·13-s − 16-s − 6·17-s + 3·18-s − 20-s − 4·22-s + 4·23-s + 25-s + 2·26-s − 2·29-s + 4·31-s − 5·32-s + 6·34-s + 3·36-s − 10·37-s + 3·40-s − 6·41-s + 4·43-s − 4·44-s − 3·45-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 9-s − 0.316·10-s + 1.20·11-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.371·29-s + 0.718·31-s − 0.883·32-s + 1.02·34-s + 1/2·36-s − 1.64·37-s + 0.474·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s − 0.447·45-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2005\)    =    \(5 \cdot 401\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2005} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2005,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;401\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;401\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 - T \)
401 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.910159619650622296165115496591, −8.344221776937260259402405455960, −7.18606324297450196513347237566, −6.58387442700360860775917772549, −5.53061415107159207269254211386, −4.73263370021136221235882375157, −3.83064612679619982339324173145, −2.57947891595731839507631448040, −1.42475002819210777825730774545, 0, 1.42475002819210777825730774545, 2.57947891595731839507631448040, 3.83064612679619982339324173145, 4.73263370021136221235882375157, 5.53061415107159207269254211386, 6.58387442700360860775917772549, 7.18606324297450196513347237566, 8.344221776937260259402405455960, 8.910159619650622296165115496591

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