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  + 2.37·2-s + 3.66·4-s − 2.55·5-s + 1.67·7-s + 3.95·8-s − 6.07·10-s − 11-s + 5.65·13-s + 3.99·14-s + 2.08·16-s + 4.01·17-s + 2.77·19-s − 9.34·20-s − 2.37·22-s − 6.13·23-s + 1.51·25-s + 13.4·26-s + 6.14·28-s − 1.25·29-s + 5.37·31-s − 2.94·32-s + 9.56·34-s − 4.27·35-s + 2.89·37-s + 6.59·38-s − 10.0·40-s − 6.31·41-s + ⋯
L(s)  = 1  + 1.68·2-s + 1.83·4-s − 1.14·5-s + 0.633·7-s + 1.39·8-s − 1.92·10-s − 0.301·11-s + 1.56·13-s + 1.06·14-s + 0.522·16-s + 0.974·17-s + 0.636·19-s − 2.08·20-s − 0.507·22-s − 1.28·23-s + 0.302·25-s + 2.63·26-s + 1.16·28-s − 0.233·29-s + 0.965·31-s − 0.519·32-s + 1.64·34-s − 0.723·35-s + 0.475·37-s + 1.07·38-s − 1.59·40-s − 0.986·41-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

\(L(1)\) \(\approx\) \(5.353530835\)
\(L(\frac12)\) \(\approx\) \(5.353530835\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.37T + 2T^{2} \)
5 \( 1 + 2.55T + 5T^{2} \)
7 \( 1 - 1.67T + 7T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 4.01T + 17T^{2} \)
19 \( 1 - 2.77T + 19T^{2} \)
23 \( 1 + 6.13T + 23T^{2} \)
29 \( 1 + 1.25T + 29T^{2} \)
31 \( 1 - 5.37T + 31T^{2} \)
37 \( 1 - 2.89T + 37T^{2} \)
41 \( 1 + 6.31T + 41T^{2} \)
43 \( 1 - 9.33T + 43T^{2} \)
47 \( 1 - 8.72T + 47T^{2} \)
53 \( 1 - 4.90T + 53T^{2} \)
59 \( 1 - 4.68T + 59T^{2} \)
67 \( 1 - 5.67T + 67T^{2} \)
71 \( 1 - 2.71T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + 1.47T + 79T^{2} \)
83 \( 1 - 2.19T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + 7.84T + 97T^{2} \)
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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

−7.982747395276507766817304431881, −7.29306149793305285814813808558, −6.39898456888438476928828405061, −5.74650160512452071556222431092, −5.16379918721726693476750313607, −4.23621001920461051648301769597, −3.83110892847491270811144066632, −3.22675530737769001949617675893, −2.18982533363954486561705855382, −0.980646825568715714633671622017, 0.980646825568715714633671622017, 2.18982533363954486561705855382, 3.22675530737769001949617675893, 3.83110892847491270811144066632, 4.23621001920461051648301769597, 5.16379918721726693476750313607, 5.74650160512452071556222431092, 6.39898456888438476928828405061, 7.29306149793305285814813808558, 7.982747395276507766817304431881

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