Properties

Label 2-6003-1.1-c1-0-135
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  − 2.60·2-s + 4.80·4-s + 3.69·5-s + 1.51·7-s − 7.32·8-s − 9.63·10-s + 4.23·11-s + 0.701·13-s − 3.94·14-s + 9.49·16-s + 2.20·17-s + 3.51·19-s + 17.7·20-s − 11.0·22-s − 23-s + 8.62·25-s − 1.82·26-s + 7.27·28-s + 29-s + 8.97·31-s − 10.1·32-s − 5.76·34-s + 5.58·35-s + 9.90·37-s − 9.17·38-s − 27.0·40-s − 8.88·41-s + ⋯
L(s)  = 1  − 1.84·2-s + 2.40·4-s + 1.65·5-s + 0.571·7-s − 2.58·8-s − 3.04·10-s + 1.27·11-s + 0.194·13-s − 1.05·14-s + 2.37·16-s + 0.535·17-s + 0.806·19-s + 3.96·20-s − 2.35·22-s − 0.208·23-s + 1.72·25-s − 0.358·26-s + 1.37·28-s + 0.185·29-s + 1.61·31-s − 1.78·32-s − 0.988·34-s + 0.943·35-s + 1.62·37-s − 1.48·38-s − 4.27·40-s − 1.38·41-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

\(L(1)\) \(\approx\) \(1.674048528\)
\(L(\frac12)\) \(\approx\) \(1.674048528\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.60T + 2T^{2} \)
5 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 - 1.51T + 7T^{2} \)
11 \( 1 - 4.23T + 11T^{2} \)
13 \( 1 - 0.701T + 13T^{2} \)
17 \( 1 - 2.20T + 17T^{2} \)
19 \( 1 - 3.51T + 19T^{2} \)
31 \( 1 - 8.97T + 31T^{2} \)
37 \( 1 - 9.90T + 37T^{2} \)
41 \( 1 + 8.88T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + 9.05T + 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 + 8.07T + 59T^{2} \)
61 \( 1 + 0.602T + 61T^{2} \)
67 \( 1 - 2.51T + 67T^{2} \)
71 \( 1 + 4.90T + 71T^{2} \)
73 \( 1 + 0.365T + 73T^{2} \)
79 \( 1 + 1.39T + 79T^{2} \)
83 \( 1 - 0.338T + 83T^{2} \)
89 \( 1 + 6.33T + 89T^{2} \)
97 \( 1 + 5.60T + 97T^{2} \)
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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

−8.242041665360535160876284548799, −7.58964563690249988042571572692, −6.67539073703304356141401293055, −6.27173004256157280348196077196, −5.62297891042131936868515007810, −4.55984945687730303567101085443, −3.14195585144800326442859734276, −2.32858053691054055337128428822, −1.42996968240422609587788722796, −1.06714151800649295137580868840, 1.06714151800649295137580868840, 1.42996968240422609587788722796, 2.32858053691054055337128428822, 3.14195585144800326442859734276, 4.55984945687730303567101085443, 5.62297891042131936868515007810, 6.27173004256157280348196077196, 6.67539073703304356141401293055, 7.58964563690249988042571572692, 8.242041665360535160876284548799

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