Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 89 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 5-s − 3·8-s − 10-s − 4·11-s − 6·13-s − 16-s + 6·17-s + 8·19-s + 20-s − 4·22-s − 8·23-s + 25-s − 6·26-s − 6·29-s − 4·31-s + 5·32-s + 6·34-s + 2·37-s + 8·38-s + 3·40-s + 6·41-s + 4·43-s + 4·44-s − 8·46-s − 7·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s − 1.20·11-s − 1.66·13-s − 1/4·16-s + 1.45·17-s + 1.83·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s − 1.17·26-s − 1.11·29-s − 0.718·31-s + 0.883·32-s + 1.02·34-s + 0.328·37-s + 1.29·38-s + 0.474·40-s + 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4005\)    =    \(3^{2} \cdot 5 \cdot 89\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4005} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4005,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.259639989$
$L(\frac12)$  $\approx$  $1.259639989$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;89\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;89\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
89 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−8.146202063001343333418383946177, −7.72599231559197505315957318438, −7.19752949399766977110725961254, −5.76330938702987298190551180645, −5.43133576226304629847483472279, −4.79928131936580810015551866294, −3.83026869442654314236078087892, −3.17039510608884563762840221305, −2.28305339484288399123535724073, −0.55475894028473866688325407921, 0.55475894028473866688325407921, 2.28305339484288399123535724073, 3.17039510608884563762840221305, 3.83026869442654314236078087892, 4.79928131936580810015551866294, 5.43133576226304629847483472279, 5.76330938702987298190551180645, 7.19752949399766977110725961254, 7.72599231559197505315957318438, 8.146202063001343333418383946177

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