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  − 2.11·2-s + 2.49·4-s − 5-s − 4.83·7-s − 1.04·8-s + 2.11·10-s + 2.21·11-s − 2.26·13-s + 10.2·14-s − 2.77·16-s + 3.10·17-s − 5.93·19-s − 2.49·20-s − 4.69·22-s − 5.79·23-s + 25-s + 4.79·26-s − 12.0·28-s − 2.32·29-s + 4.47·31-s + 7.96·32-s − 6.59·34-s + 4.83·35-s − 8.23·37-s + 12.5·38-s + 1.04·40-s + 0.278·41-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.24·4-s − 0.447·5-s − 1.82·7-s − 0.369·8-s + 0.670·10-s + 0.668·11-s − 0.627·13-s + 2.73·14-s − 0.692·16-s + 0.754·17-s − 1.36·19-s − 0.557·20-s − 1.00·22-s − 1.20·23-s + 0.200·25-s + 0.940·26-s − 2.27·28-s − 0.432·29-s + 0.803·31-s + 1.40·32-s − 1.13·34-s + 0.816·35-s − 1.35·37-s + 2.04·38-s + 0.165·40-s + 0.0434·41-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  =  $(2,\ 4005,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2045254584$
$L(\frac12)$  $\approx$  $0.2045254584$
$L(\frac{3}{2})$   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 \( 1 + 2.11T + 2T^{2} \)
7 \( 1 + 4.83T + 7T^{2} \)
11 \( 1 - 2.21T + 11T^{2} \)
13 \( 1 + 2.26T + 13T^{2} \)
17 \( 1 - 3.10T + 17T^{2} \)
19 \( 1 + 5.93T + 19T^{2} \)
23 \( 1 + 5.79T + 23T^{2} \)
29 \( 1 + 2.32T + 29T^{2} \)
31 \( 1 - 4.47T + 31T^{2} \)
37 \( 1 + 8.23T + 37T^{2} \)
41 \( 1 - 0.278T + 41T^{2} \)
43 \( 1 + 0.176T + 43T^{2} \)
47 \( 1 + 1.91T + 47T^{2} \)
53 \( 1 + 8.65T + 53T^{2} \)
59 \( 1 + 5.39T + 59T^{2} \)
61 \( 1 + 9.69T + 61T^{2} \)
67 \( 1 - 8.06T + 67T^{2} \)
71 \( 1 - 5.00T + 71T^{2} \)
73 \( 1 + 9.18T + 73T^{2} \)
79 \( 1 + 1.78T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
97 \( 1 + 5.52T + 97T^{2} \)
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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

−8.543780460767788644017514108180, −7.86570926182645227720502383751, −7.07191460419984150102116424347, −6.54854560279542335399131599390, −5.90561765764338513550873879229, −4.50506348867043270778594110557, −3.67030964179931905986957882141, −2.78843777062835554305536232888, −1.70020202082051440673147507162, −0.31640057740750242824242205185, 0.31640057740750242824242205185, 1.70020202082051440673147507162, 2.78843777062835554305536232888, 3.67030964179931905986957882141, 4.50506348867043270778594110557, 5.90561765764338513550873879229, 6.54854560279542335399131599390, 7.07191460419984150102116424347, 7.86570926182645227720502383751, 8.543780460767788644017514108180

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