Properties

Degree 2
Conductor $ 3^{2} \cdot 23 \cdot 29 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.16·2-s − 0.638·4-s − 1.66·5-s − 0.970·7-s − 3.07·8-s − 1.94·10-s + 3.13·11-s + 3.36·13-s − 1.13·14-s − 2.31·16-s − 3.46·17-s + 3.42·19-s + 1.06·20-s + 3.65·22-s + 23-s − 2.23·25-s + 3.92·26-s + 0.619·28-s + 29-s − 6.67·31-s + 3.45·32-s − 4.04·34-s + 1.61·35-s − 3.48·37-s + 3.99·38-s + 5.12·40-s + 7.11·41-s + ⋯
L(s)  = 1  + 0.825·2-s − 0.319·4-s − 0.744·5-s − 0.366·7-s − 1.08·8-s − 0.614·10-s + 0.943·11-s + 0.933·13-s − 0.302·14-s − 0.578·16-s − 0.840·17-s + 0.785·19-s + 0.237·20-s + 0.778·22-s + 0.208·23-s − 0.446·25-s + 0.770·26-s + 0.117·28-s + 0.185·29-s − 1.19·31-s + 0.610·32-s − 0.693·34-s + 0.273·35-s − 0.572·37-s + 0.648·38-s + 0.810·40-s + 1.11·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6003} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6003,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
23 \( 1 - T \)
29 \( 1 - T \)
good2 \( 1 - 1.16T + 2T^{2} \)
5 \( 1 + 1.66T + 5T^{2} \)
7 \( 1 + 0.970T + 7T^{2} \)
11 \( 1 - 3.13T + 11T^{2} \)
13 \( 1 - 3.36T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 3.42T + 19T^{2} \)
31 \( 1 + 6.67T + 31T^{2} \)
37 \( 1 + 3.48T + 37T^{2} \)
41 \( 1 - 7.11T + 41T^{2} \)
43 \( 1 - 5.93T + 43T^{2} \)
47 \( 1 - 6.42T + 47T^{2} \)
53 \( 1 - 8.21T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + 1.90T + 61T^{2} \)
67 \( 1 + 0.345T + 67T^{2} \)
71 \( 1 + 9.30T + 71T^{2} \)
73 \( 1 - 2.86T + 73T^{2} \)
79 \( 1 - 3.43T + 79T^{2} \)
83 \( 1 - 1.04T + 83T^{2} \)
89 \( 1 + 9.73T + 89T^{2} \)
97 \( 1 + 8.02T + 97T^{2} \)
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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

−7.64130089738963484201874461486, −6.90570028535916511187710430986, −6.13465031993285335739048943940, −5.60979681524552037028774797067, −4.63340272022332008654253224231, −3.93891406249969218150124667164, −3.61378910585857994573400198823, −2.66609552357521578210166966653, −1.25913244539314970333823178733, 0, 1.25913244539314970333823178733, 2.66609552357521578210166966653, 3.61378910585857994573400198823, 3.93891406249969218150124667164, 4.63340272022332008654253224231, 5.60979681524552037028774797067, 6.13465031993285335739048943940, 6.90570028535916511187710430986, 7.64130089738963484201874461486

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