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  + 1.14·2-s − 0.686·4-s + 0.461·5-s − 3.03·7-s − 3.07·8-s + 0.528·10-s + 1.24·11-s + 3.02·13-s − 3.47·14-s − 2.15·16-s + 1.56·17-s − 0.758·19-s − 0.316·20-s + 1.42·22-s + 23-s − 4.78·25-s + 3.47·26-s + 2.08·28-s + 29-s + 5.58·31-s + 3.68·32-s + 1.79·34-s − 1.39·35-s + 11.3·37-s − 0.869·38-s − 1.42·40-s + 4.22·41-s + ⋯
L(s)  = 1  + 0.810·2-s − 0.343·4-s + 0.206·5-s − 1.14·7-s − 1.08·8-s + 0.167·10-s + 0.373·11-s + 0.839·13-s − 0.928·14-s − 0.539·16-s + 0.379·17-s − 0.174·19-s − 0.0707·20-s + 0.303·22-s + 0.208·23-s − 0.957·25-s + 0.680·26-s + 0.393·28-s + 0.185·29-s + 1.00·31-s + 0.651·32-s + 0.307·34-s − 0.236·35-s + 1.87·37-s − 0.141·38-s − 0.224·40-s + 0.659·41-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 \( 1 - 1.14T + 2T^{2} \)
5 \( 1 - 0.461T + 5T^{2} \)
7 \( 1 + 3.03T + 7T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 - 3.02T + 13T^{2} \)
17 \( 1 - 1.56T + 17T^{2} \)
19 \( 1 + 0.758T + 19T^{2} \)
31 \( 1 - 5.58T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 - 4.22T + 41T^{2} \)
43 \( 1 + 3.44T + 43T^{2} \)
47 \( 1 + 13.0T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 1.58T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 4.97T + 67T^{2} \)
71 \( 1 - 3.12T + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 - 5.95T + 79T^{2} \)
83 \( 1 - 1.97T + 83T^{2} \)
89 \( 1 + 3.14T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−7.79891452708078001142631911311, −6.53919424879216258864571513385, −6.28500716880266583133166120231, −5.68562328077247972593153685514, −4.70138619487039718388658803761, −4.06940758366657401543106373968, −3.29993722412638097101714282663, −2.75557047375670208099649049727, −1.30720716999374041212564945384, 0, 1.30720716999374041212564945384, 2.75557047375670208099649049727, 3.29993722412638097101714282663, 4.06940758366657401543106373968, 4.70138619487039718388658803761, 5.68562328077247972593153685514, 6.28500716880266583133166120231, 6.53919424879216258864571513385, 7.79891452708078001142631911311

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