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  + 0.144·2-s − 1.97·4-s − 4.08·5-s − 0.828·7-s − 0.574·8-s − 0.590·10-s + 0.0328·11-s − 3.12·13-s − 0.119·14-s + 3.87·16-s + 1.27·17-s − 4.11·19-s + 8.09·20-s + 0.00474·22-s + 23-s + 11.7·25-s − 0.452·26-s + 1.63·28-s + 29-s + 6.85·31-s + 1.70·32-s + 0.183·34-s + 3.38·35-s − 3.30·37-s − 0.595·38-s + 2.35·40-s + 6.64·41-s + ⋯
L(s)  = 1  + 0.102·2-s − 0.989·4-s − 1.82·5-s − 0.313·7-s − 0.203·8-s − 0.186·10-s + 0.00991·11-s − 0.868·13-s − 0.0319·14-s + 0.968·16-s + 0.308·17-s − 0.945·19-s + 1.80·20-s + 0.00101·22-s + 0.208·23-s + 2.34·25-s − 0.0886·26-s + 0.309·28-s + 0.185·29-s + 1.23·31-s + 0.302·32-s + 0.0314·34-s + 0.572·35-s − 0.542·37-s − 0.0965·38-s + 0.371·40-s + 1.03·41-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 \( 1 - 0.144T + 2T^{2} \)
5 \( 1 + 4.08T + 5T^{2} \)
7 \( 1 + 0.828T + 7T^{2} \)
11 \( 1 - 0.0328T + 11T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 - 1.27T + 17T^{2} \)
19 \( 1 + 4.11T + 19T^{2} \)
31 \( 1 - 6.85T + 31T^{2} \)
37 \( 1 + 3.30T + 37T^{2} \)
41 \( 1 - 6.64T + 41T^{2} \)
43 \( 1 + 1.68T + 43T^{2} \)
47 \( 1 - 5.69T + 47T^{2} \)
53 \( 1 - 7.40T + 53T^{2} \)
59 \( 1 + 2.70T + 59T^{2} \)
61 \( 1 - 7.51T + 61T^{2} \)
67 \( 1 + 4.02T + 67T^{2} \)
71 \( 1 + 3.73T + 71T^{2} \)
73 \( 1 - 2.01T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + 7.36T + 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−7.87216568231884167634778803415, −7.16336218112022253216554700190, −6.41770708283532483731678285393, −5.33043676867115594818084153861, −4.61695281489698523808938079593, −4.12676004932706355134996166731, −3.45023289563605861629796634050, −2.61634960698360727317209060096, −0.866377066549047050069742269284, 0, 0.866377066549047050069742269284, 2.61634960698360727317209060096, 3.45023289563605861629796634050, 4.12676004932706355134996166731, 4.61695281489698523808938079593, 5.33043676867115594818084153861, 6.41770708283532483731678285393, 7.16336218112022253216554700190, 7.87216568231884167634778803415

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