Properties

Label 2-6003-1.1-c1-0-248
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  + 1.96·2-s + 1.84·4-s + 2.84·5-s − 2.21·7-s − 0.305·8-s + 5.57·10-s − 1.29·11-s − 0.673·13-s − 4.34·14-s − 4.28·16-s − 1.19·17-s − 5.61·19-s + 5.24·20-s − 2.54·22-s − 23-s + 3.07·25-s − 1.32·26-s − 4.08·28-s − 29-s − 4.05·31-s − 7.79·32-s − 2.33·34-s − 6.29·35-s + 3.39·37-s − 11.0·38-s − 0.867·40-s − 9.19·41-s + ⋯
L(s)  = 1  + 1.38·2-s + 0.922·4-s + 1.27·5-s − 0.837·7-s − 0.107·8-s + 1.76·10-s − 0.390·11-s − 0.186·13-s − 1.16·14-s − 1.07·16-s − 0.289·17-s − 1.28·19-s + 1.17·20-s − 0.542·22-s − 0.208·23-s + 0.614·25-s − 0.259·26-s − 0.771·28-s − 0.185·29-s − 0.728·31-s − 1.37·32-s − 0.401·34-s − 1.06·35-s + 0.557·37-s − 1.78·38-s − 0.137·40-s − 1.43·41-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 \( 1 - 1.96T + 2T^{2} \)
5 \( 1 - 2.84T + 5T^{2} \)
7 \( 1 + 2.21T + 7T^{2} \)
11 \( 1 + 1.29T + 11T^{2} \)
13 \( 1 + 0.673T + 13T^{2} \)
17 \( 1 + 1.19T + 17T^{2} \)
19 \( 1 + 5.61T + 19T^{2} \)
31 \( 1 + 4.05T + 31T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 + 9.19T + 41T^{2} \)
43 \( 1 - 9.58T + 43T^{2} \)
47 \( 1 - 4.53T + 47T^{2} \)
53 \( 1 + 3.31T + 53T^{2} \)
59 \( 1 + 7.04T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 - 4.14T + 67T^{2} \)
71 \( 1 - 4.40T + 71T^{2} \)
73 \( 1 + 6.12T + 73T^{2} \)
79 \( 1 - 1.11T + 79T^{2} \)
83 \( 1 - 2.93T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 - 0.608T + 97T^{2} \)
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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

−7.41524143443643729208403393060, −6.59004578612467188975911434796, −6.10965846233448450311812596413, −5.66000487612556978672180833894, −4.84293656485232704519493963959, −4.15567074409747772382927630646, −3.25704745065001960089858159648, −2.52222704338575769145325275146, −1.84182565668712350139677905046, 0, 1.84182565668712350139677905046, 2.52222704338575769145325275146, 3.25704745065001960089858159648, 4.15567074409747772382927630646, 4.84293656485232704519493963959, 5.66000487612556978672180833894, 6.10965846233448450311812596413, 6.59004578612467188975911434796, 7.41524143443643729208403393060

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