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  + 1.83·2-s + 1.37·4-s − 5-s − 4.72·7-s − 1.14·8-s − 1.83·10-s + 4.62·11-s − 4.47·13-s − 8.68·14-s − 4.85·16-s + 5.66·17-s + 1.34·19-s − 1.37·20-s + 8.50·22-s − 4.44·23-s + 25-s − 8.23·26-s − 6.50·28-s + 4.27·29-s + 9.40·31-s − 6.63·32-s + 10.4·34-s + 4.72·35-s + 6.89·37-s + 2.46·38-s + 1.14·40-s − 5.69·41-s + ⋯
L(s)  = 1  + 1.29·2-s + 0.688·4-s − 0.447·5-s − 1.78·7-s − 0.404·8-s − 0.581·10-s + 1.39·11-s − 1.24·13-s − 2.32·14-s − 1.21·16-s + 1.37·17-s + 0.308·19-s − 0.307·20-s + 1.81·22-s − 0.926·23-s + 0.200·25-s − 1.61·26-s − 1.23·28-s + 0.793·29-s + 1.68·31-s − 1.17·32-s + 1.78·34-s + 0.799·35-s + 1.13·37-s + 0.400·38-s + 0.180·40-s − 0.889·41-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  =  $(2,\ 4005,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.434538851$
$L(\frac12)$  $\approx$  $2.434538851$
$L(\frac{3}{2})$   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 \( 1 - 1.83T + 2T^{2} \)
7 \( 1 + 4.72T + 7T^{2} \)
11 \( 1 - 4.62T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 5.66T + 17T^{2} \)
19 \( 1 - 1.34T + 19T^{2} \)
23 \( 1 + 4.44T + 23T^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 - 9.40T + 31T^{2} \)
37 \( 1 - 6.89T + 37T^{2} \)
41 \( 1 + 5.69T + 41T^{2} \)
43 \( 1 + 5.04T + 43T^{2} \)
47 \( 1 - 8.76T + 47T^{2} \)
53 \( 1 - 8.23T + 53T^{2} \)
59 \( 1 - 7.52T + 59T^{2} \)
61 \( 1 + 6.94T + 61T^{2} \)
67 \( 1 + 6.38T + 67T^{2} \)
71 \( 1 + 5.84T + 71T^{2} \)
73 \( 1 - 11.0T + 73T^{2} \)
79 \( 1 - 7.89T + 79T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
97 \( 1 + 0.674T + 97T^{2} \)
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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

−8.424362546112816198535620504993, −7.41458420987855150610434395871, −6.65016456159708239310275815663, −6.24856046148251180478569870096, −5.46183766801109674851122371644, −4.50573107891205291556732860319, −3.83958330471190222236067231831, −3.22152493562563395571105655414, −2.53960951102450828506834178394, −0.71471803262329141204066186509, 0.71471803262329141204066186509, 2.53960951102450828506834178394, 3.22152493562563395571105655414, 3.83958330471190222236067231831, 4.50573107891205291556732860319, 5.46183766801109674851122371644, 6.24856046148251180478569870096, 6.65016456159708239310275815663, 7.41458420987855150610434395871, 8.424362546112816198535620504993

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