Properties

Label 2-6003-1.1-c1-0-10
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  − 2.10·2-s + 2.44·4-s − 1.70·5-s − 4.14·7-s − 0.943·8-s + 3.58·10-s + 1.69·11-s − 1.39·13-s + 8.74·14-s − 2.90·16-s + 0.447·17-s − 1.61·19-s − 4.16·20-s − 3.56·22-s − 23-s − 2.10·25-s + 2.93·26-s − 10.1·28-s − 29-s − 9.36·31-s + 8.01·32-s − 0.943·34-s + 7.05·35-s + 4.24·37-s + 3.41·38-s + 1.60·40-s − 6.78·41-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.22·4-s − 0.760·5-s − 1.56·7-s − 0.333·8-s + 1.13·10-s + 0.509·11-s − 0.385·13-s + 2.33·14-s − 0.726·16-s + 0.108·17-s − 0.371·19-s − 0.930·20-s − 0.760·22-s − 0.208·23-s − 0.421·25-s + 0.575·26-s − 1.91·28-s − 0.185·29-s − 1.68·31-s + 1.41·32-s − 0.161·34-s + 1.19·35-s + 0.697·37-s + 0.553·38-s + 0.253·40-s − 1.05·41-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

\(L(1)\) \(\approx\) \(0.1402966145\)
\(L(\frac12)\) \(\approx\) \(0.1402966145\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.10T + 2T^{2} \)
5 \( 1 + 1.70T + 5T^{2} \)
7 \( 1 + 4.14T + 7T^{2} \)
11 \( 1 - 1.69T + 11T^{2} \)
13 \( 1 + 1.39T + 13T^{2} \)
17 \( 1 - 0.447T + 17T^{2} \)
19 \( 1 + 1.61T + 19T^{2} \)
31 \( 1 + 9.36T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + 6.78T + 41T^{2} \)
43 \( 1 + 8.81T + 43T^{2} \)
47 \( 1 - 5.76T + 47T^{2} \)
53 \( 1 - 3.27T + 53T^{2} \)
59 \( 1 + 3.19T + 59T^{2} \)
61 \( 1 + 6.01T + 61T^{2} \)
67 \( 1 + 0.789T + 67T^{2} \)
71 \( 1 - 4.82T + 71T^{2} \)
73 \( 1 - 1.62T + 73T^{2} \)
79 \( 1 - 2.87T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 - 1.11T + 89T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−8.116034315489907200443513360094, −7.50345324596294166140959906009, −6.86725965577489407012607936525, −6.37859626751802682159635263451, −5.38555251819046093284439038050, −4.16379948005214299826485468050, −3.57786336273429890426853870673, −2.61828340321222722327521360351, −1.55640174135404040268195725499, −0.24655011630279649231392011666, 0.24655011630279649231392011666, 1.55640174135404040268195725499, 2.61828340321222722327521360351, 3.57786336273429890426853870673, 4.16379948005214299826485468050, 5.38555251819046093284439038050, 6.37859626751802682159635263451, 6.86725965577489407012607936525, 7.50345324596294166140959906009, 8.116034315489907200443513360094

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