Properties

Label 2-6003-1.1-c1-0-213
Degree $2$
Conductor $6003$
Sign $-1$
Analytic cond. $47.9341$
Root an. cond. $6.92345$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.424·2-s − 1.81·4-s + 2.43·5-s + 3.41·7-s − 1.62·8-s + 1.03·10-s − 5.84·11-s − 6.73·13-s + 1.44·14-s + 2.95·16-s + 4.57·17-s + 3.36·19-s − 4.43·20-s − 2.47·22-s − 23-s + 0.941·25-s − 2.85·26-s − 6.21·28-s − 29-s − 2.25·31-s + 4.49·32-s + 1.94·34-s + 8.31·35-s − 4.84·37-s + 1.42·38-s − 3.95·40-s + 4.80·41-s + ⋯
L(s)  = 1  + 0.300·2-s − 0.909·4-s + 1.09·5-s + 1.28·7-s − 0.573·8-s + 0.327·10-s − 1.76·11-s − 1.86·13-s + 0.387·14-s + 0.737·16-s + 1.11·17-s + 0.772·19-s − 0.991·20-s − 0.528·22-s − 0.208·23-s + 0.188·25-s − 0.560·26-s − 1.17·28-s − 0.185·29-s − 0.405·31-s + 0.794·32-s + 0.333·34-s + 1.40·35-s − 0.795·37-s + 0.231·38-s − 0.624·40-s + 0.749·41-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: $-1$
Analytic conductor: \(47.9341\)
Root analytic conductor: \(6.92345\)
Motivic weight: \(1\)
Rational: no
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 \( 1 - 0.424T + 2T^{2} \)
5 \( 1 - 2.43T + 5T^{2} \)
7 \( 1 - 3.41T + 7T^{2} \)
11 \( 1 + 5.84T + 11T^{2} \)
13 \( 1 + 6.73T + 13T^{2} \)
17 \( 1 - 4.57T + 17T^{2} \)
19 \( 1 - 3.36T + 19T^{2} \)
31 \( 1 + 2.25T + 31T^{2} \)
37 \( 1 + 4.84T + 37T^{2} \)
41 \( 1 - 4.80T + 41T^{2} \)
43 \( 1 - 7.09T + 43T^{2} \)
47 \( 1 - 3.50T + 47T^{2} \)
53 \( 1 + 0.803T + 53T^{2} \)
59 \( 1 + 8.37T + 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 - 7.03T + 67T^{2} \)
71 \( 1 + 6.87T + 71T^{2} \)
73 \( 1 + 3.29T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + 7.50T + 89T^{2} \)
97 \( 1 + 9.65T + 97T^{2} \)
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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

−7.59343315881127110036450546510, −7.42407374569621831638263358297, −5.70269704610749981386319859501, −5.57913223416676504282198837876, −4.94236639853783089442879577014, −4.40194410018639790884475452237, −3.04849669890227720149634178572, −2.41837383574939413767301300342, −1.42123009106035484415693663512, 0, 1.42123009106035484415693663512, 2.41837383574939413767301300342, 3.04849669890227720149634178572, 4.40194410018639790884475452237, 4.94236639853783089442879577014, 5.57913223416676504282198837876, 5.70269704610749981386319859501, 7.42407374569621831638263358297, 7.59343315881127110036450546510

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