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  + 1.66·2-s + 0.765·4-s + 5-s − 1.19·7-s − 2.05·8-s + 1.66·10-s + 1.71·11-s − 6.71·13-s − 1.98·14-s − 4.94·16-s − 0.198·17-s + 5.88·19-s + 0.765·20-s + 2.85·22-s + 5.52·23-s + 25-s − 11.1·26-s − 0.915·28-s − 9.07·29-s + 2.91·31-s − 4.11·32-s − 0.329·34-s − 1.19·35-s + 1.00·37-s + 9.78·38-s − 2.05·40-s − 9.69·41-s + ⋯
L(s)  = 1  + 1.17·2-s + 0.382·4-s + 0.447·5-s − 0.451·7-s − 0.725·8-s + 0.525·10-s + 0.517·11-s − 1.86·13-s − 0.531·14-s − 1.23·16-s − 0.0480·17-s + 1.35·19-s + 0.171·20-s + 0.608·22-s + 1.15·23-s + 0.200·25-s − 2.19·26-s − 0.172·28-s − 1.68·29-s + 0.522·31-s − 0.727·32-s − 0.0565·34-s − 0.202·35-s + 0.164·37-s + 1.58·38-s − 0.324·40-s − 1.51·41-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 \( 1 - 1.66T + 2T^{2} \)
7 \( 1 + 1.19T + 7T^{2} \)
11 \( 1 - 1.71T + 11T^{2} \)
13 \( 1 + 6.71T + 13T^{2} \)
17 \( 1 + 0.198T + 17T^{2} \)
19 \( 1 - 5.88T + 19T^{2} \)
23 \( 1 - 5.52T + 23T^{2} \)
29 \( 1 + 9.07T + 29T^{2} \)
31 \( 1 - 2.91T + 31T^{2} \)
37 \( 1 - 1.00T + 37T^{2} \)
41 \( 1 + 9.69T + 41T^{2} \)
43 \( 1 + 6.27T + 43T^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 + 8.93T + 53T^{2} \)
59 \( 1 + 3.20T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 15.0T + 67T^{2} \)
71 \( 1 + 3.52T + 71T^{2} \)
73 \( 1 + 3.36T + 73T^{2} \)
79 \( 1 + 5.21T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−7.917027593670105814455487190026, −7.01143728695841323739143354034, −6.59373242825626213253764533477, −5.54039935810480456619635497232, −5.11401240897820509304204560388, −4.44058491491209292530569847480, −3.30238711904642034757048576045, −2.91239416510549673016004835419, −1.70465997737705327945372653031, 0, 1.70465997737705327945372653031, 2.91239416510549673016004835419, 3.30238711904642034757048576045, 4.44058491491209292530569847480, 5.11401240897820509304204560388, 5.54039935810480456619635497232, 6.59373242825626213253764533477, 7.01143728695841323739143354034, 7.917027593670105814455487190026

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