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  + 0.948·2-s − 1.10·4-s + 2.25·5-s + 5.24·7-s − 2.94·8-s + 2.13·10-s − 11-s + 5.44·13-s + 4.96·14-s − 0.586·16-s − 4.46·17-s + 2.98·19-s − 2.47·20-s − 0.948·22-s − 0.200·23-s + 0.0679·25-s + 5.15·26-s − 5.76·28-s + 1.81·29-s + 0.728·31-s + 5.32·32-s − 4.22·34-s + 11.7·35-s − 3.44·37-s + 2.83·38-s − 6.61·40-s − 2.51·41-s + ⋯
L(s)  = 1  + 0.670·2-s − 0.550·4-s + 1.00·5-s + 1.98·7-s − 1.03·8-s + 0.675·10-s − 0.301·11-s + 1.50·13-s + 1.32·14-s − 0.146·16-s − 1.08·17-s + 0.685·19-s − 0.554·20-s − 0.202·22-s − 0.0417·23-s + 0.0135·25-s + 1.01·26-s − 1.09·28-s + 0.337·29-s + 0.130·31-s + 0.941·32-s − 0.725·34-s + 1.99·35-s − 0.566·37-s + 0.459·38-s − 1.04·40-s − 0.392·41-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

\(L(1)\) \(\approx\) \(3.888463469\)
\(L(\frac12)\) \(\approx\) \(3.888463469\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.948T + 2T^{2} \)
5 \( 1 - 2.25T + 5T^{2} \)
7 \( 1 - 5.24T + 7T^{2} \)
13 \( 1 - 5.44T + 13T^{2} \)
17 \( 1 + 4.46T + 17T^{2} \)
19 \( 1 - 2.98T + 19T^{2} \)
23 \( 1 + 0.200T + 23T^{2} \)
29 \( 1 - 1.81T + 29T^{2} \)
31 \( 1 - 0.728T + 31T^{2} \)
37 \( 1 + 3.44T + 37T^{2} \)
41 \( 1 + 2.51T + 41T^{2} \)
43 \( 1 - 7.72T + 43T^{2} \)
47 \( 1 - 9.29T + 47T^{2} \)
53 \( 1 - 1.48T + 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
67 \( 1 - 16.1T + 67T^{2} \)
71 \( 1 + 1.39T + 71T^{2} \)
73 \( 1 + 8.30T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 1.10T + 83T^{2} \)
89 \( 1 + 6.80T + 89T^{2} \)
97 \( 1 + 9.30T + 97T^{2} \)
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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

−8.314043418298803019137226517137, −7.38189810493732077714791701551, −6.39715209750468171363386153161, −5.56594605790249926491283454462, −5.37330343057999113206004322033, −4.42213821699670005800516669485, −3.98364292684918082047146231046, −2.75330326642730690653899548196, −1.86587476640607329099681152317, −1.02388964045873747986552837651, 1.02388964045873747986552837651, 1.86587476640607329099681152317, 2.75330326642730690653899548196, 3.98364292684918082047146231046, 4.42213821699670005800516669485, 5.37330343057999113206004322033, 5.56594605790249926491283454462, 6.39715209750468171363386153161, 7.38189810493732077714791701551, 8.314043418298803019137226517137

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