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  + 1.05·2-s − 0.877·4-s + 1.30·5-s + 0.720·7-s − 3.04·8-s + 1.38·10-s − 4.26·11-s + 6.49·13-s + 0.763·14-s − 1.47·16-s − 1.60·17-s + 0.0676·19-s − 1.14·20-s − 4.52·22-s − 23-s − 3.29·25-s + 6.88·26-s − 0.632·28-s + 29-s − 3.80·31-s + 4.53·32-s − 1.70·34-s + 0.939·35-s − 3.80·37-s + 0.0716·38-s − 3.97·40-s + 0.467·41-s + ⋯
L(s)  = 1  + 0.749·2-s − 0.438·4-s + 0.583·5-s + 0.272·7-s − 1.07·8-s + 0.436·10-s − 1.28·11-s + 1.80·13-s + 0.204·14-s − 0.369·16-s − 0.389·17-s + 0.0155·19-s − 0.255·20-s − 0.964·22-s − 0.208·23-s − 0.659·25-s + 1.34·26-s − 0.119·28-s + 0.185·29-s − 0.682·31-s + 0.801·32-s − 0.292·34-s + 0.158·35-s − 0.624·37-s + 0.0116·38-s − 0.628·40-s + 0.0730·41-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 \( 1 - 1.05T + 2T^{2} \)
5 \( 1 - 1.30T + 5T^{2} \)
7 \( 1 - 0.720T + 7T^{2} \)
11 \( 1 + 4.26T + 11T^{2} \)
13 \( 1 - 6.49T + 13T^{2} \)
17 \( 1 + 1.60T + 17T^{2} \)
19 \( 1 - 0.0676T + 19T^{2} \)
31 \( 1 + 3.80T + 31T^{2} \)
37 \( 1 + 3.80T + 37T^{2} \)
41 \( 1 - 0.467T + 41T^{2} \)
43 \( 1 - 0.971T + 43T^{2} \)
47 \( 1 + 7.97T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 - 4.90T + 59T^{2} \)
61 \( 1 - 7.29T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 0.843T + 71T^{2} \)
73 \( 1 + 3.56T + 73T^{2} \)
79 \( 1 - 3.50T + 79T^{2} \)
83 \( 1 - 8.90T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 0.847T + 97T^{2} \)
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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

−8.036048221167308348690597656174, −6.70204514412249048582571332440, −6.14919360156160235323595743359, −5.41496713566803163899786998798, −5.01277948400919531768370425861, −4.00654664682682474916560513151, −3.42541868020511158802717292950, −2.47641730216395080393316494665, −1.45662568297017411635798652251, 0, 1.45662568297017411635798652251, 2.47641730216395080393316494665, 3.42541868020511158802717292950, 4.00654664682682474916560513151, 5.01277948400919531768370425861, 5.41496713566803163899786998798, 6.14919360156160235323595743359, 6.70204514412249048582571332440, 8.036048221167308348690597656174

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