Properties

Label 2-6003-1.1-c1-0-42
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.27·2-s + 3.18·4-s + 2.98·5-s − 2.44·7-s − 2.70·8-s − 6.79·10-s − 3.22·11-s − 2.73·13-s + 5.57·14-s − 0.220·16-s − 2.14·17-s + 1.46·19-s + 9.50·20-s + 7.34·22-s + 23-s + 3.90·25-s + 6.22·26-s − 7.79·28-s − 29-s + 6.98·31-s + 5.90·32-s + 4.87·34-s − 7.30·35-s + 10.5·37-s − 3.34·38-s − 8.06·40-s − 6.90·41-s + ⋯
L(s)  = 1  − 1.61·2-s + 1.59·4-s + 1.33·5-s − 0.924·7-s − 0.955·8-s − 2.14·10-s − 0.972·11-s − 0.757·13-s + 1.48·14-s − 0.0551·16-s − 0.519·17-s + 0.336·19-s + 2.12·20-s + 1.56·22-s + 0.208·23-s + 0.781·25-s + 1.21·26-s − 1.47·28-s − 0.185·29-s + 1.25·31-s + 1.04·32-s + 0.836·34-s − 1.23·35-s + 1.73·37-s − 0.542·38-s − 1.27·40-s − 1.07·41-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: \(0\)
Selberg data: \((2,\ 6003,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6812803205\)
\(L(\frac12)\) \(\approx\) \(0.6812803205\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.27T + 2T^{2} \)
5 \( 1 - 2.98T + 5T^{2} \)
7 \( 1 + 2.44T + 7T^{2} \)
11 \( 1 + 3.22T + 11T^{2} \)
13 \( 1 + 2.73T + 13T^{2} \)
17 \( 1 + 2.14T + 17T^{2} \)
19 \( 1 - 1.46T + 19T^{2} \)
31 \( 1 - 6.98T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + 6.90T + 41T^{2} \)
43 \( 1 + 9.50T + 43T^{2} \)
47 \( 1 + 3.28T + 47T^{2} \)
53 \( 1 + 7.20T + 53T^{2} \)
59 \( 1 - 7.50T + 59T^{2} \)
61 \( 1 + 0.996T + 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 + 5.24T + 71T^{2} \)
73 \( 1 - 8.24T + 73T^{2} \)
79 \( 1 + 7.48T + 79T^{2} \)
83 \( 1 + 7.33T + 83T^{2} \)
89 \( 1 - 7.50T + 89T^{2} \)
97 \( 1 - 8.72T + 97T^{2} \)
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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

−8.195700607854632245806457483260, −7.54436324534547327313795538655, −6.64711351985719013837018825143, −6.37614042224701054716548606485, −5.38996134557264362500564966212, −4.65397874201562907778759483880, −3.10139630666940662340778738087, −2.47443877579885311215072125644, −1.74714351386224145405821349410, −0.53735737997428305345268284130, 0.53735737997428305345268284130, 1.74714351386224145405821349410, 2.47443877579885311215072125644, 3.10139630666940662340778738087, 4.65397874201562907778759483880, 5.38996134557264362500564966212, 6.37614042224701054716548606485, 6.64711351985719013837018825143, 7.54436324534547327313795538655, 8.195700607854632245806457483260

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