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 + 4·11-s + 3·13-s + 4·16-s − 3·17-s − 7·19-s − 23-s − 5·25-s + 29-s + 2·31-s + 37-s + 6·41-s − 11·43-s − 8·44-s − 10·47-s − 7·49-s − 6·52-s + 6·53-s + 11·59-s + 14·61-s − 8·64-s + 6·68-s + 15·71-s − 12·73-s + 14·76-s + 79-s + 2·83-s + ⋯
L(s)  = 1  − 4-s + 1.20·11-s + 0.832·13-s + 16-s − 0.727·17-s − 1.60·19-s − 0.208·23-s − 25-s + 0.185·29-s + 0.359·31-s + 0.164·37-s + 0.937·41-s − 1.67·43-s − 1.20·44-s − 1.45·47-s − 49-s − 0.832·52-s + 0.824·53-s + 1.43·59-s + 1.79·61-s − 64-s + 0.727·68-s + 1.78·71-s − 1.40·73-s + 1.60·76-s + 0.112·79-s + 0.219·83-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 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.072951861163183414204383866975, −6.76777130438189262938336518541, −6.41098223414322409427369702225, −5.58762817816198073910432983640, −4.66003404692236358902664260720, −4.01933909460077403178405683484, −3.58255753087140901376892284759, −2.21796908624490588502252820857, −1.23855514836103712898121634803, 0, 1.23855514836103712898121634803, 2.21796908624490588502252820857, 3.58255753087140901376892284759, 4.01933909460077403178405683484, 4.66003404692236358902664260720, 5.58762817816198073910432983640, 6.41098223414322409427369702225, 6.76777130438189262938336518541, 8.072951861163183414204383866975

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