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  − 1.17·2-s − 0.615·4-s − 2.85·5-s + 3.62·7-s + 3.07·8-s + 3.35·10-s − 1.39·11-s + 2.03·13-s − 4.27·14-s − 2.38·16-s + 0.256·17-s + 1.59·19-s + 1.75·20-s + 1.64·22-s − 23-s + 3.14·25-s − 2.39·26-s − 2.23·28-s + 29-s + 3.05·31-s − 3.34·32-s − 0.301·34-s − 10.3·35-s − 5.80·37-s − 1.87·38-s − 8.78·40-s − 10.7·41-s + ⋯
L(s)  = 1  − 0.831·2-s − 0.307·4-s − 1.27·5-s + 1.37·7-s + 1.08·8-s + 1.06·10-s − 0.421·11-s + 0.564·13-s − 1.14·14-s − 0.597·16-s + 0.0622·17-s + 0.365·19-s + 0.392·20-s + 0.351·22-s − 0.208·23-s + 0.628·25-s − 0.470·26-s − 0.422·28-s + 0.185·29-s + 0.548·31-s − 0.590·32-s − 0.0517·34-s − 1.75·35-s − 0.954·37-s − 0.304·38-s − 1.38·40-s − 1.68·41-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

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 \( 1 + 1.17T + 2T^{2} \)
5 \( 1 + 2.85T + 5T^{2} \)
7 \( 1 - 3.62T + 7T^{2} \)
11 \( 1 + 1.39T + 11T^{2} \)
13 \( 1 - 2.03T + 13T^{2} \)
17 \( 1 - 0.256T + 17T^{2} \)
19 \( 1 - 1.59T + 19T^{2} \)
31 \( 1 - 3.05T + 31T^{2} \)
37 \( 1 + 5.80T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 9.74T + 53T^{2} \)
59 \( 1 - 4.00T + 59T^{2} \)
61 \( 1 - 7.96T + 61T^{2} \)
67 \( 1 - 1.95T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 8.14T + 73T^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + 6.27T + 83T^{2} \)
89 \( 1 + 3.67T + 89T^{2} \)
97 \( 1 - 8.10T + 97T^{2} \)
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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

−8.052888207743698067611302320390, −7.33642640176240535918477753048, −6.63326217080195831939508651498, −5.15836996058350391698929451216, −4.98736506762546297989469867244, −4.01677065968762297244011873974, −3.42251600979811956468981086376, −1.98066885143951786097632502445, −1.11166536771958539267378662538, 0, 1.11166536771958539267378662538, 1.98066885143951786097632502445, 3.42251600979811956468981086376, 4.01677065968762297244011873974, 4.98736506762546297989469867244, 5.15836996058350391698929451216, 6.63326217080195831939508651498, 7.33642640176240535918477753048, 8.052888207743698067611302320390

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