Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.11·2-s − 0.753·4-s + 5-s − 1.16·7-s − 3.07·8-s + 1.11·10-s + 2.19·11-s + 1.10·13-s − 1.30·14-s − 1.92·16-s − 7.78·17-s + 4.75·19-s − 0.753·20-s + 2.44·22-s − 2.26·23-s + 25-s + 1.22·26-s + 0.879·28-s + 6.01·29-s − 1.58·31-s + 3.99·32-s − 8.68·34-s − 1.16·35-s − 7.29·37-s + 5.30·38-s − 3.07·40-s + 2.30·41-s + ⋯
L(s)  = 1  + 0.789·2-s − 0.376·4-s + 0.447·5-s − 0.441·7-s − 1.08·8-s + 0.353·10-s + 0.660·11-s + 0.305·13-s − 0.348·14-s − 0.481·16-s − 1.88·17-s + 1.08·19-s − 0.168·20-s + 0.521·22-s − 0.473·23-s + 0.200·25-s + 0.241·26-s + 0.166·28-s + 1.11·29-s − 0.284·31-s + 0.706·32-s − 1.48·34-s − 0.197·35-s − 1.19·37-s + 0.860·38-s − 0.486·40-s + 0.360·41-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  =  1
Selberg data  =  $(2,\ 4005,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;89\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;89\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 - T \)
89 \( 1 + T \)
good2 \( 1 - 1.11T + 2T^{2} \)
7 \( 1 + 1.16T + 7T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 - 1.10T + 13T^{2} \)
17 \( 1 + 7.78T + 17T^{2} \)
19 \( 1 - 4.75T + 19T^{2} \)
23 \( 1 + 2.26T + 23T^{2} \)
29 \( 1 - 6.01T + 29T^{2} \)
31 \( 1 + 1.58T + 31T^{2} \)
37 \( 1 + 7.29T + 37T^{2} \)
41 \( 1 - 2.30T + 41T^{2} \)
43 \( 1 + 2.34T + 43T^{2} \)
47 \( 1 - 0.949T + 47T^{2} \)
53 \( 1 + 8.03T + 53T^{2} \)
59 \( 1 + 3.82T + 59T^{2} \)
61 \( 1 + 5.72T + 61T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 + 2.59T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 3.60T + 83T^{2} \)
97 \( 1 + 3.17T + 97T^{2} \)
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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

−8.216860420941867203403418920126, −7.06883487658277976193197346672, −6.38660131586960686664033885595, −5.91097552334956955894040600739, −4.91717407377683495665100466755, −4.38366902062929512246306769110, −3.48495091435878125806561972854, −2.76046409449729015729018324750, −1.52985507192588514682833156098, 0, 1.52985507192588514682833156098, 2.76046409449729015729018324750, 3.48495091435878125806561972854, 4.38366902062929512246306769110, 4.91717407377683495665100466755, 5.91097552334956955894040600739, 6.38660131586960686664033885595, 7.06883487658277976193197346672, 8.216860420941867203403418920126

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