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  + 2.59·2-s + 4.75·4-s + 3.50·5-s + 0.252·7-s + 7.15·8-s + 9.12·10-s − 11-s + 0.314·13-s + 0.655·14-s + 9.09·16-s − 3.13·17-s + 0.841·19-s + 16.6·20-s − 2.59·22-s − 1.45·23-s + 7.31·25-s + 0.818·26-s + 1.19·28-s + 6.76·29-s − 7.07·31-s + 9.31·32-s − 8.14·34-s + 0.885·35-s + 8.89·37-s + 2.18·38-s + 25.1·40-s + 8.43·41-s + ⋯
L(s)  = 1  + 1.83·2-s + 2.37·4-s + 1.56·5-s + 0.0953·7-s + 2.53·8-s + 2.88·10-s − 0.301·11-s + 0.0873·13-s + 0.175·14-s + 2.27·16-s − 0.759·17-s + 0.192·19-s + 3.73·20-s − 0.554·22-s − 0.302·23-s + 1.46·25-s + 0.160·26-s + 0.226·28-s + 1.25·29-s − 1.27·31-s + 1.64·32-s − 1.39·34-s + 0.149·35-s + 1.46·37-s + 0.354·38-s + 3.97·40-s + 1.31·41-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

\(L(1)\) \(\approx\) \(9.201506065\)
\(L(\frac12)\) \(\approx\) \(9.201506065\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.59T + 2T^{2} \)
5 \( 1 - 3.50T + 5T^{2} \)
7 \( 1 - 0.252T + 7T^{2} \)
13 \( 1 - 0.314T + 13T^{2} \)
17 \( 1 + 3.13T + 17T^{2} \)
19 \( 1 - 0.841T + 19T^{2} \)
23 \( 1 + 1.45T + 23T^{2} \)
29 \( 1 - 6.76T + 29T^{2} \)
31 \( 1 + 7.07T + 31T^{2} \)
37 \( 1 - 8.89T + 37T^{2} \)
41 \( 1 - 8.43T + 41T^{2} \)
43 \( 1 - 1.80T + 43T^{2} \)
47 \( 1 + 2.30T + 47T^{2} \)
53 \( 1 - 4.41T + 53T^{2} \)
59 \( 1 + 4.48T + 59T^{2} \)
67 \( 1 + 0.0414T + 67T^{2} \)
71 \( 1 - 9.80T + 71T^{2} \)
73 \( 1 - 8.90T + 73T^{2} \)
79 \( 1 + 8.51T + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 + 5.58T + 89T^{2} \)
97 \( 1 - 7.54T + 97T^{2} \)
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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

−7.76150331778775917890710578827, −6.97599412090788720707800763192, −6.21873702851072677606500537954, −5.94801506529834795680864603008, −5.15707993674438100248816556304, −4.62160089579703320433531979987, −3.76900866065674534459902329138, −2.70536996686098970975922819483, −2.31898912895677229430092213381, −1.37425451499831577401015566464, 1.37425451499831577401015566464, 2.31898912895677229430092213381, 2.70536996686098970975922819483, 3.76900866065674534459902329138, 4.62160089579703320433531979987, 5.15707993674438100248816556304, 5.94801506529834795680864603008, 6.21873702851072677606500537954, 6.97599412090788720707800763192, 7.76150331778775917890710578827

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