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  − 0.230·2-s − 1.94·4-s + 2.59·5-s − 4.77·7-s + 0.908·8-s − 0.597·10-s − 3.70·11-s − 3.06·13-s + 1.09·14-s + 3.68·16-s + 3.40·17-s + 5.00·19-s − 5.05·20-s + 0.851·22-s + 23-s + 1.73·25-s + 0.705·26-s + 9.30·28-s + 29-s + 6.71·31-s − 2.66·32-s − 0.783·34-s − 12.3·35-s + 1.21·37-s − 1.15·38-s + 2.35·40-s − 8.51·41-s + ⋯
L(s)  = 1  − 0.162·2-s − 0.973·4-s + 1.16·5-s − 1.80·7-s + 0.321·8-s − 0.188·10-s − 1.11·11-s − 0.850·13-s + 0.293·14-s + 0.921·16-s + 0.825·17-s + 1.14·19-s − 1.12·20-s + 0.181·22-s + 0.208·23-s + 0.346·25-s + 0.138·26-s + 1.75·28-s + 0.185·29-s + 1.20·31-s − 0.471·32-s − 0.134·34-s − 2.09·35-s + 0.200·37-s − 0.186·38-s + 0.372·40-s − 1.33·41-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 \( 1 + 0.230T + 2T^{2} \)
5 \( 1 - 2.59T + 5T^{2} \)
7 \( 1 + 4.77T + 7T^{2} \)
11 \( 1 + 3.70T + 11T^{2} \)
13 \( 1 + 3.06T + 13T^{2} \)
17 \( 1 - 3.40T + 17T^{2} \)
19 \( 1 - 5.00T + 19T^{2} \)
31 \( 1 - 6.71T + 31T^{2} \)
37 \( 1 - 1.21T + 37T^{2} \)
41 \( 1 + 8.51T + 41T^{2} \)
43 \( 1 - 8.34T + 43T^{2} \)
47 \( 1 - 3.13T + 47T^{2} \)
53 \( 1 - 6.13T + 53T^{2} \)
59 \( 1 + 3.83T + 59T^{2} \)
61 \( 1 - 3.39T + 61T^{2} \)
67 \( 1 - 1.42T + 67T^{2} \)
71 \( 1 + 1.21T + 71T^{2} \)
73 \( 1 + 8.66T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 1.64T + 97T^{2} \)
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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

−7.69442494261824225621559709576, −7.06449921069340640502924227262, −6.15272939110937072006388485654, −5.52580473772572177876564086146, −5.10331350303714354902108684919, −4.00377659408005950211101951723, −3.01913353448513183393632434313, −2.60176955670822698713086660857, −1.09416265554881086901775926429, 0, 1.09416265554881086901775926429, 2.60176955670822698713086660857, 3.01913353448513183393632434313, 4.00377659408005950211101951723, 5.10331350303714354902108684919, 5.52580473772572177876564086146, 6.15272939110937072006388485654, 7.06449921069340640502924227262, 7.69442494261824225621559709576

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