Properties

Degree $2$
Conductor $6039$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.46·2-s + 0.154·4-s − 3.84·5-s + 2.46·7-s − 2.70·8-s − 5.64·10-s − 11-s − 4.64·13-s + 3.61·14-s − 4.28·16-s − 5.70·17-s + 6.71·19-s − 0.594·20-s − 1.46·22-s − 2.93·23-s + 9.76·25-s − 6.82·26-s + 0.381·28-s − 5.67·29-s + 0.232·31-s − 0.874·32-s − 8.37·34-s − 9.46·35-s + 5.55·37-s + 9.85·38-s + 10.4·40-s + 4.35·41-s + ⋯
L(s)  = 1  + 1.03·2-s + 0.0774·4-s − 1.71·5-s + 0.930·7-s − 0.957·8-s − 1.78·10-s − 0.301·11-s − 1.28·13-s + 0.965·14-s − 1.07·16-s − 1.38·17-s + 1.53·19-s − 0.133·20-s − 0.312·22-s − 0.612·23-s + 1.95·25-s − 1.33·26-s + 0.0720·28-s − 1.05·29-s + 0.0417·31-s − 0.154·32-s − 1.43·34-s − 1.59·35-s + 0.912·37-s + 1.59·38-s + 1.64·40-s + 0.680·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6039} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.266085115\)
\(L(\frac12)\) \(\approx\) \(1.266085115\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + T \)
61 \( 1 + T \)
good2 \( 1 - 1.46T + 2T^{2} \)
5 \( 1 + 3.84T + 5T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
13 \( 1 + 4.64T + 13T^{2} \)
17 \( 1 + 5.70T + 17T^{2} \)
19 \( 1 - 6.71T + 19T^{2} \)
23 \( 1 + 2.93T + 23T^{2} \)
29 \( 1 + 5.67T + 29T^{2} \)
31 \( 1 - 0.232T + 31T^{2} \)
37 \( 1 - 5.55T + 37T^{2} \)
41 \( 1 - 4.35T + 41T^{2} \)
43 \( 1 + 9.35T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 - 0.299T + 71T^{2} \)
73 \( 1 + 2.65T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 7.42T + 83T^{2} \)
89 \( 1 - 3.40T + 89T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−8.022905784675026991812276360126, −7.32736928586743365135050235270, −6.79644190145913840437622181079, −5.57143178146289976955954786081, −4.95574733148785872949731921913, −4.48565674430134741828393592194, −3.85030570135814113276858260392, −3.08819759185783626126966260949, −2.16149840244085666868950148157, −0.48212482915960218693547648457, 0.48212482915960218693547648457, 2.16149840244085666868950148157, 3.08819759185783626126966260949, 3.85030570135814113276858260392, 4.48565674430134741828393592194, 4.95574733148785872949731921913, 5.57143178146289976955954786081, 6.79644190145913840437622181079, 7.32736928586743365135050235270, 8.022905784675026991812276360126

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