Properties

Degree $2$
Conductor $6003$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 2.44·5-s + 0.449·7-s + 13-s + 4·16-s − 3.44·17-s − 19-s − 4.89·20-s + 23-s + 0.999·25-s − 0.898·28-s − 29-s − 5.55·31-s + 1.10·35-s + 0.101·37-s − 5.34·41-s − 9.89·43-s + 7.55·47-s − 6.79·49-s − 2·52-s + 2·53-s + 5.44·59-s − 12.8·61-s − 8·64-s + 2.44·65-s + 0.898·67-s + 6.89·68-s + ⋯
L(s)  = 1  − 4-s + 1.09·5-s + 0.169·7-s + 0.277·13-s + 16-s − 0.836·17-s − 0.229·19-s − 1.09·20-s + 0.208·23-s + 0.199·25-s − 0.169·28-s − 0.185·29-s − 0.996·31-s + 0.186·35-s + 0.0166·37-s − 0.835·41-s − 1.50·43-s + 1.10·47-s − 0.971·49-s − 0.277·52-s + 0.274·53-s + 0.709·59-s − 1.65·61-s − 64-s + 0.303·65-s + 0.109·67-s + 0.836·68-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

Degree: \(2\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6003} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6003,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 - T \)
29 \( 1 + T \)
good2 \( 1 + 2T^{2} \)
5 \( 1 - 2.44T + 5T^{2} \)
7 \( 1 - 0.449T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + 3.44T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
31 \( 1 + 5.55T + 31T^{2} \)
37 \( 1 - 0.101T + 37T^{2} \)
41 \( 1 + 5.34T + 41T^{2} \)
43 \( 1 + 9.89T + 43T^{2} \)
47 \( 1 - 7.55T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 5.44T + 59T^{2} \)
61 \( 1 + 12.8T + 61T^{2} \)
67 \( 1 - 0.898T + 67T^{2} \)
71 \( 1 + 2.34T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + 3.89T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + 0.550T + 89T^{2} \)
97 \( 1 - 7.10T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.897096936263980881127084495039, −6.90747393217435094851690655451, −6.22653860577511542911024161343, −5.46902720402960450624228798244, −4.95646128454021496260441118014, −4.11443465569389324016325618141, −3.31272655702183652625344964858, −2.18581143826740635576365838203, −1.38517596150965033471377151376, 0, 1.38517596150965033471377151376, 2.18581143826740635576365838203, 3.31272655702183652625344964858, 4.11443465569389324016325618141, 4.95646128454021496260441118014, 5.46902720402960450624228798244, 6.22653860577511542911024161343, 6.90747393217435094851690655451, 7.897096936263980881127084495039

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