Properties

Label 2-6003-1.1-c1-0-104
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  − 0.603·2-s − 1.63·4-s + 2.98·5-s + 4.14·7-s + 2.19·8-s − 1.80·10-s + 1.78·11-s − 3.75·13-s − 2.50·14-s + 1.94·16-s − 3.76·17-s + 2.44·19-s − 4.88·20-s − 1.08·22-s − 23-s + 3.90·25-s + 2.26·26-s − 6.77·28-s − 29-s − 0.393·31-s − 5.56·32-s + 2.27·34-s + 12.3·35-s − 0.656·37-s − 1.47·38-s + 6.55·40-s + 7.56·41-s + ⋯
L(s)  = 1  − 0.427·2-s − 0.817·4-s + 1.33·5-s + 1.56·7-s + 0.776·8-s − 0.570·10-s + 0.539·11-s − 1.04·13-s − 0.668·14-s + 0.486·16-s − 0.912·17-s + 0.560·19-s − 1.09·20-s − 0.230·22-s − 0.208·23-s + 0.781·25-s + 0.444·26-s − 1.27·28-s − 0.185·29-s − 0.0707·31-s − 0.983·32-s + 0.389·34-s + 2.08·35-s − 0.107·37-s − 0.239·38-s + 1.03·40-s + 1.18·41-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

\(L(1)\) \(\approx\) \(2.099402099\)
\(L(\frac12)\) \(\approx\) \(2.099402099\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.603T + 2T^{2} \)
5 \( 1 - 2.98T + 5T^{2} \)
7 \( 1 - 4.14T + 7T^{2} \)
11 \( 1 - 1.78T + 11T^{2} \)
13 \( 1 + 3.75T + 13T^{2} \)
17 \( 1 + 3.76T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
31 \( 1 + 0.393T + 31T^{2} \)
37 \( 1 + 0.656T + 37T^{2} \)
41 \( 1 - 7.56T + 41T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 + 7.63T + 53T^{2} \)
59 \( 1 - 4.61T + 59T^{2} \)
61 \( 1 - 2.70T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 - 2.39T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 9.97T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + 0.336T + 89T^{2} \)
97 \( 1 - 8.25T + 97T^{2} \)
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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

−8.095884528589636644071803685850, −7.61540753867093082964939937176, −6.70480919772554865934551685808, −5.81211626356069623981114351152, −5.04816548303586333958025900504, −4.73451901530020608428294829948, −3.82342078402134787211852858524, −2.36238010600036818268310934311, −1.81349931888512112484689471248, −0.863315600727074405770138933069, 0.863315600727074405770138933069, 1.81349931888512112484689471248, 2.36238010600036818268310934311, 3.82342078402134787211852858524, 4.73451901530020608428294829948, 5.04816548303586333958025900504, 5.81211626356069623981114351152, 6.70480919772554865934551685808, 7.61540753867093082964939937176, 8.095884528589636644071803685850

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