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  − 2.51·2-s + 4.32·4-s + 5-s − 2.83·7-s − 5.84·8-s − 2.51·10-s + 1.32·11-s − 5.12·13-s + 7.11·14-s + 6.04·16-s − 4.61·17-s + 1.84·19-s + 4.32·20-s − 3.34·22-s + 2.29·23-s + 25-s + 12.8·26-s − 12.2·28-s − 0.167·29-s + 9.85·31-s − 3.51·32-s + 11.5·34-s − 2.83·35-s − 0.207·37-s − 4.64·38-s − 5.84·40-s − 5.20·41-s + ⋯
L(s)  = 1  − 1.77·2-s + 2.16·4-s + 0.447·5-s − 1.06·7-s − 2.06·8-s − 0.795·10-s + 0.400·11-s − 1.42·13-s + 1.90·14-s + 1.51·16-s − 1.11·17-s + 0.423·19-s + 0.966·20-s − 0.712·22-s + 0.477·23-s + 0.200·25-s + 2.52·26-s − 2.31·28-s − 0.0310·29-s + 1.76·31-s − 0.622·32-s + 1.98·34-s − 0.478·35-s − 0.0341·37-s − 0.753·38-s − 0.923·40-s − 0.812·41-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 \( 1 + 2.51T + 2T^{2} \)
7 \( 1 + 2.83T + 7T^{2} \)
11 \( 1 - 1.32T + 11T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 + 4.61T + 17T^{2} \)
19 \( 1 - 1.84T + 19T^{2} \)
23 \( 1 - 2.29T + 23T^{2} \)
29 \( 1 + 0.167T + 29T^{2} \)
31 \( 1 - 9.85T + 31T^{2} \)
37 \( 1 + 0.207T + 37T^{2} \)
41 \( 1 + 5.20T + 41T^{2} \)
43 \( 1 - 8.88T + 43T^{2} \)
47 \( 1 - 5.91T + 47T^{2} \)
53 \( 1 - 3.11T + 53T^{2} \)
59 \( 1 - 7.20T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 - 9.69T + 67T^{2} \)
71 \( 1 + 0.991T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 8.00T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
97 \( 1 - 1.80T + 97T^{2} \)
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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

−8.306272688669127696664357610112, −7.31829655384236861795594683182, −6.86133356013752849553906966846, −6.33915629412473082872898439942, −5.32497909513918770934889478380, −4.17800599529065440761723392238, −2.77588875433909425430119883925, −2.40029136759666425083287819783, −1.09918351396455551656916745411, 0, 1.09918351396455551656916745411, 2.40029136759666425083287819783, 2.77588875433909425430119883925, 4.17800599529065440761723392238, 5.32497909513918770934889478380, 6.33915629412473082872898439942, 6.86133356013752849553906966846, 7.31829655384236861795594683182, 8.306272688669127696664357610112

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