Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 41 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.80·2-s + 2.24·4-s + 7-s + 2.24·8-s − 1.80·13-s + 1.80·14-s + 1.80·16-s + 0.445·17-s − 0.445·19-s − 1.24·23-s + 25-s − 3.24·26-s + 2.24·28-s + 1.00·32-s + 0.801·34-s − 1.80·37-s − 0.801·38-s − 41-s − 0.445·43-s − 2.24·46-s + 1.80·47-s + 49-s + 1.80·50-s − 4.04·52-s + 2.24·56-s + 68-s − 3.24·74-s + ⋯
L(s)  = 1  + 1.80·2-s + 2.24·4-s + 7-s + 2.24·8-s − 1.80·13-s + 1.80·14-s + 1.80·16-s + 0.445·17-s − 0.445·19-s − 1.24·23-s + 25-s − 3.24·26-s + 2.24·28-s + 1.00·32-s + 0.801·34-s − 1.80·37-s − 0.801·38-s − 41-s − 0.445·43-s − 2.24·46-s + 1.80·47-s + 49-s + 1.80·50-s − 4.04·52-s + 2.24·56-s + 68-s − 3.24·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2583\)    =    \(3^{2} \cdot 7 \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2583} (2008, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2583,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(3.602451674\)
\(L(\frac12)\)  \(\approx\)  \(3.602451674\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;41\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
41 \( 1 + T \)
good2 \( 1 - 1.80T + T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.80T + T^{2} \)
17 \( 1 - 0.445T + T^{2} \)
19 \( 1 + 0.445T + T^{2} \)
23 \( 1 + 1.24T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.80T + T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 - 1.80T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.80T + T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−8.983000295474648906364810549000, −7.976857762952515399555409853895, −7.28914788626878081825581172147, −6.66009108189749755568573142615, −5.59944926380598088895231367307, −5.05962152005061891764338708651, −4.48618270099382737373914540532, −3.59931104148218534627880623277, −2.54489260032120497780155444424, −1.83975269323596061561585091536, 1.83975269323596061561585091536, 2.54489260032120497780155444424, 3.59931104148218534627880623277, 4.48618270099382737373914540532, 5.05962152005061891764338708651, 5.59944926380598088895231367307, 6.66009108189749755568573142615, 7.28914788626878081825581172147, 7.976857762952515399555409853895, 8.983000295474648906364810549000

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