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Properties

Label 2-6003-1.1-c1-0-71
Degree $2$
Conductor $6003$
Sign $-1$
Analytic cond. $47.9341$
Root an. cond. $6.92345$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 4·5-s − 4·7-s − 4·11-s − 5·13-s + 4·16-s + 5·17-s + 5·19-s + 8·20-s − 23-s + 11·25-s + 8·28-s + 29-s − 2·31-s + 16·35-s + 5·37-s + 2·41-s + 43-s + 8·44-s − 6·47-s + 9·49-s + 10·52-s − 2·53-s + 16·55-s − 9·59-s − 10·61-s − 8·64-s + ⋯
L(s)  = 1  − 4-s − 1.78·5-s − 1.51·7-s − 1.20·11-s − 1.38·13-s + 16-s + 1.21·17-s + 1.14·19-s + 1.78·20-s − 0.208·23-s + 11/5·25-s + 1.51·28-s + 0.185·29-s − 0.359·31-s + 2.70·35-s + 0.821·37-s + 0.312·41-s + 0.152·43-s + 1.20·44-s − 0.875·47-s + 9/7·49-s + 1.38·52-s − 0.274·53-s + 2.15·55-s − 1.17·59-s − 1.28·61-s − 64-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: $-1$
Analytic conductor: \(47.9341\)
Root analytic conductor: \(6.92345\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.71563520178021378819704348045, −7.39943062113386048368553094077, −6.35175977066125987796095421757, −5.28989831531475286198018533996, −4.87500219121284891311108676008, −3.92916553362581132715595761506, −3.28027512672173009491613515873, −2.82911129067469526207624771855, −0.71010362222413778805237105454, 0, 0.71010362222413778805237105454, 2.82911129067469526207624771855, 3.28027512672173009491613515873, 3.92916553362581132715595761506, 4.87500219121284891311108676008, 5.28989831531475286198018533996, 6.35175977066125987796095421757, 7.39943062113386048368553094077, 7.71563520178021378819704348045

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