Properties

Label 2-6003-1.1-c1-0-114
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  + 0.480·2-s − 1.76·4-s + 1.51·5-s + 3.82·7-s − 1.81·8-s + 0.726·10-s + 4.55·11-s − 2.03·13-s + 1.84·14-s + 2.66·16-s + 3.09·17-s − 5.07·19-s − 2.67·20-s + 2.19·22-s + 23-s − 2.71·25-s − 0.977·26-s − 6.77·28-s − 29-s + 3.30·31-s + 4.90·32-s + 1.48·34-s + 5.78·35-s + 9.63·37-s − 2.44·38-s − 2.73·40-s + 7.83·41-s + ⋯
L(s)  = 1  + 0.340·2-s − 0.884·4-s + 0.675·5-s + 1.44·7-s − 0.640·8-s + 0.229·10-s + 1.37·11-s − 0.563·13-s + 0.492·14-s + 0.666·16-s + 0.750·17-s − 1.16·19-s − 0.597·20-s + 0.467·22-s + 0.208·23-s − 0.543·25-s − 0.191·26-s − 1.28·28-s − 0.185·29-s + 0.594·31-s + 0.867·32-s + 0.255·34-s + 0.978·35-s + 1.58·37-s − 0.395·38-s − 0.433·40-s + 1.22·41-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

\(L(1)\) \(\approx\) \(2.844241815\)
\(L(\frac12)\) \(\approx\) \(2.844241815\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.480T + 2T^{2} \)
5 \( 1 - 1.51T + 5T^{2} \)
7 \( 1 - 3.82T + 7T^{2} \)
11 \( 1 - 4.55T + 11T^{2} \)
13 \( 1 + 2.03T + 13T^{2} \)
17 \( 1 - 3.09T + 17T^{2} \)
19 \( 1 + 5.07T + 19T^{2} \)
31 \( 1 - 3.30T + 31T^{2} \)
37 \( 1 - 9.63T + 37T^{2} \)
41 \( 1 - 7.83T + 41T^{2} \)
43 \( 1 - 1.74T + 43T^{2} \)
47 \( 1 - 6.78T + 47T^{2} \)
53 \( 1 + 8.59T + 53T^{2} \)
59 \( 1 - 2.86T + 59T^{2} \)
61 \( 1 + 3.14T + 61T^{2} \)
67 \( 1 + 3.13T + 67T^{2} \)
71 \( 1 + 5.22T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 + 2.01T + 79T^{2} \)
83 \( 1 - 0.169T + 83T^{2} \)
89 \( 1 + 4.44T + 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−8.099638054482893610291778005055, −7.54649555422246743290604417367, −6.41656514580658278674889975881, −5.85759918459982195403480068111, −5.14894646259507665458740250220, −4.33151355963132304030506013723, −4.06470581399440927782845408093, −2.75747861696066302371213394136, −1.77490624990787483168288493960, −0.912792947891547208975010876264, 0.912792947891547208975010876264, 1.77490624990787483168288493960, 2.75747861696066302371213394136, 4.06470581399440927782845408093, 4.33151355963132304030506013723, 5.14894646259507665458740250220, 5.85759918459982195403480068111, 6.41656514580658278674889975881, 7.54649555422246743290604417367, 8.099638054482893610291778005055

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