Properties

Label 2-6003-1.1-c1-0-120
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  − 1.20·2-s − 0.559·4-s + 3.40·5-s + 4.71·7-s + 3.07·8-s − 4.08·10-s + 1.68·11-s − 2.33·13-s − 5.65·14-s − 2.56·16-s + 4.01·17-s + 0.666·19-s − 1.90·20-s − 2.02·22-s − 23-s + 6.56·25-s + 2.80·26-s − 2.63·28-s + 29-s − 4.38·31-s − 3.06·32-s − 4.82·34-s + 16.0·35-s + 1.15·37-s − 0.799·38-s + 10.4·40-s − 10.3·41-s + ⋯
L(s)  = 1  − 0.848·2-s − 0.279·4-s + 1.52·5-s + 1.78·7-s + 1.08·8-s − 1.29·10-s + 0.508·11-s − 0.647·13-s − 1.51·14-s − 0.641·16-s + 0.974·17-s + 0.152·19-s − 0.425·20-s − 0.431·22-s − 0.208·23-s + 1.31·25-s + 0.549·26-s − 0.498·28-s + 0.185·29-s − 0.787·31-s − 0.541·32-s − 0.826·34-s + 2.70·35-s + 0.189·37-s − 0.129·38-s + 1.65·40-s − 1.61·41-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

\(L(1)\) \(\approx\) \(2.139252688\)
\(L(\frac12)\) \(\approx\) \(2.139252688\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.20T + 2T^{2} \)
5 \( 1 - 3.40T + 5T^{2} \)
7 \( 1 - 4.71T + 7T^{2} \)
11 \( 1 - 1.68T + 11T^{2} \)
13 \( 1 + 2.33T + 13T^{2} \)
17 \( 1 - 4.01T + 17T^{2} \)
19 \( 1 - 0.666T + 19T^{2} \)
31 \( 1 + 4.38T + 31T^{2} \)
37 \( 1 - 1.15T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 4.96T + 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 + 7.60T + 53T^{2} \)
59 \( 1 + 2.67T + 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 - 7.64T + 67T^{2} \)
71 \( 1 - 2.23T + 71T^{2} \)
73 \( 1 - 3.18T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 0.691T + 83T^{2} \)
89 \( 1 - 18.4T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−8.171255552650849976908583369674, −7.58686276278185413819890440827, −6.86198399775542412172890018943, −5.79840585729662181128448774763, −5.16109517205134644231195978966, −4.76530839606022147072469804684, −3.68878763896000204630180369162, −2.24310017539424520863876629531, −1.70238692860697245978621194745, −0.973611190490846696448294376533, 0.973611190490846696448294376533, 1.70238692860697245978621194745, 2.24310017539424520863876629531, 3.68878763896000204630180369162, 4.76530839606022147072469804684, 5.16109517205134644231195978966, 5.79840585729662181128448774763, 6.86198399775542412172890018943, 7.58686276278185413819890440827, 8.171255552650849976908583369674

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