Properties

Label 2-6003-1.1-c1-0-21
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.395·2-s − 1.84·4-s − 0.579·5-s − 2.59·7-s − 1.52·8-s − 0.229·10-s + 1.87·11-s − 6.20·13-s − 1.02·14-s + 3.08·16-s + 4.01·17-s − 2.79·19-s + 1.06·20-s + 0.742·22-s + 23-s − 4.66·25-s − 2.45·26-s + 4.78·28-s − 29-s − 3.08·31-s + 4.26·32-s + 1.58·34-s + 1.50·35-s − 3.22·37-s − 1.10·38-s + 0.881·40-s − 8.40·41-s + ⋯
L(s)  = 1  + 0.279·2-s − 0.921·4-s − 0.259·5-s − 0.981·7-s − 0.537·8-s − 0.0725·10-s + 0.565·11-s − 1.72·13-s − 0.274·14-s + 0.770·16-s + 0.973·17-s − 0.640·19-s + 0.238·20-s + 0.158·22-s + 0.208·23-s − 0.932·25-s − 0.482·26-s + 0.904·28-s − 0.185·29-s − 0.554·31-s + 0.753·32-s + 0.272·34-s + 0.254·35-s − 0.530·37-s − 0.179·38-s + 0.139·40-s − 1.31·41-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

\(L(1)\) \(\approx\) \(0.6268022284\)
\(L(\frac12)\) \(\approx\) \(0.6268022284\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.395T + 2T^{2} \)
5 \( 1 + 0.579T + 5T^{2} \)
7 \( 1 + 2.59T + 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 + 6.20T + 13T^{2} \)
17 \( 1 - 4.01T + 17T^{2} \)
19 \( 1 + 2.79T + 19T^{2} \)
31 \( 1 + 3.08T + 31T^{2} \)
37 \( 1 + 3.22T + 37T^{2} \)
41 \( 1 + 8.40T + 41T^{2} \)
43 \( 1 + 6.56T + 43T^{2} \)
47 \( 1 + 0.0783T + 47T^{2} \)
53 \( 1 - 9.39T + 53T^{2} \)
59 \( 1 + 7.94T + 59T^{2} \)
61 \( 1 + 9.00T + 61T^{2} \)
67 \( 1 - 0.596T + 67T^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 - 0.703T + 73T^{2} \)
79 \( 1 - 2.70T + 79T^{2} \)
83 \( 1 + 9.68T + 83T^{2} \)
89 \( 1 - 9.96T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.052075702787218445665733453699, −7.37447225431319506005961602846, −6.65160502122293037211742294865, −5.84579410971284113528770174916, −5.13153145560195100909744936368, −4.46388582754206014657241939478, −3.59134523078849501450180452292, −3.12289611661123656681141198985, −1.89338522760801859574892917504, −0.38573716689033850972915075956, 0.38573716689033850972915075956, 1.89338522760801859574892917504, 3.12289611661123656681141198985, 3.59134523078849501450180452292, 4.46388582754206014657241939478, 5.13153145560195100909744936368, 5.84579410971284113528770174916, 6.65160502122293037211742294865, 7.37447225431319506005961602846, 8.052075702787218445665733453699

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