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 − 4.44·7-s + 13-s + 4·16-s + 1.44·17-s − 19-s + 4.89·20-s + 23-s + 0.999·25-s + 8.89·28-s − 29-s − 10.4·31-s + 10.8·35-s + 9.89·37-s + 9.34·41-s − 0.101·43-s + 12.4·47-s + 12.7·49-s − 2·52-s + 2·53-s + 0.550·59-s − 3.10·61-s − 8·64-s − 2.44·65-s − 8.89·67-s − 2.89·68-s + ⋯
L(s)  = 1  − 4-s − 1.09·5-s − 1.68·7-s + 0.277·13-s + 16-s + 0.351·17-s − 0.229·19-s + 1.09·20-s + 0.208·23-s + 0.199·25-s + 1.68·28-s − 0.185·29-s − 1.87·31-s + 1.84·35-s + 1.62·37-s + 1.45·41-s − 0.0154·43-s + 1.81·47-s + 1.82·49-s − 0.277·52-s + 0.274·53-s + 0.0716·59-s − 0.397·61-s − 64-s − 0.303·65-s − 1.08·67-s − 0.351·68-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 \( 1 + 4.44T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 - 1.44T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 - 9.89T + 37T^{2} \)
41 \( 1 - 9.34T + 41T^{2} \)
43 \( 1 + 0.101T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 0.550T + 59T^{2} \)
61 \( 1 + 3.10T + 61T^{2} \)
67 \( 1 + 8.89T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 - 5.89T + 79T^{2} \)
83 \( 1 - 7.55T + 83T^{2} \)
89 \( 1 + 5.44T + 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−7.57382637482461060688925434005, −7.30725985450537149508723027077, −6.11849428646037861499601062937, −5.74802905672627806791710638245, −4.62696712134164529345462278958, −3.83251270554898348963212431355, −3.57824233798080791223310775096, −2.59424369458930025707687990159, −0.862305715804540496833711459923, 0, 0.862305715804540496833711459923, 2.59424369458930025707687990159, 3.57824233798080791223310775096, 3.83251270554898348963212431355, 4.62696712134164529345462278958, 5.74802905672627806791710638245, 6.11849428646037861499601062937, 7.30725985450537149508723027077, 7.57382637482461060688925434005

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