Properties

Label 2-6039-1.1-c1-0-161
Degree $2$
Conductor $6039$
Sign $1$
Analytic cond. $48.2216$
Root an. cond. $6.94418$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.77·2-s + 5.72·4-s − 2.25·5-s + 0.879·7-s + 10.3·8-s − 6.27·10-s + 11-s + 3.02·13-s + 2.44·14-s + 17.2·16-s − 2.27·17-s + 3.93·19-s − 12.9·20-s + 2.77·22-s + 3.56·23-s + 0.0920·25-s + 8.40·26-s + 5.02·28-s + 1.57·29-s − 9.17·31-s + 27.3·32-s − 6.32·34-s − 1.98·35-s − 7.15·37-s + 10.9·38-s − 23.3·40-s + 7.02·41-s + ⋯
L(s)  = 1  + 1.96·2-s + 2.86·4-s − 1.00·5-s + 0.332·7-s + 3.65·8-s − 1.98·10-s + 0.301·11-s + 0.838·13-s + 0.652·14-s + 4.32·16-s − 0.552·17-s + 0.903·19-s − 2.88·20-s + 0.592·22-s + 0.743·23-s + 0.0184·25-s + 1.64·26-s + 0.950·28-s + 0.292·29-s − 1.64·31-s + 4.83·32-s − 1.08·34-s − 0.335·35-s − 1.17·37-s + 1.77·38-s − 3.68·40-s + 1.09·41-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: $1$
Analytic conductor: \(48.2216\)
Root analytic conductor: \(6.94418\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.609431119\)
\(L(\frac12)\) \(\approx\) \(7.609431119\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.77T + 2T^{2} \)
5 \( 1 + 2.25T + 5T^{2} \)
7 \( 1 - 0.879T + 7T^{2} \)
13 \( 1 - 3.02T + 13T^{2} \)
17 \( 1 + 2.27T + 17T^{2} \)
19 \( 1 - 3.93T + 19T^{2} \)
23 \( 1 - 3.56T + 23T^{2} \)
29 \( 1 - 1.57T + 29T^{2} \)
31 \( 1 + 9.17T + 31T^{2} \)
37 \( 1 + 7.15T + 37T^{2} \)
41 \( 1 - 7.02T + 41T^{2} \)
43 \( 1 - 8.40T + 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 1.97T + 53T^{2} \)
59 \( 1 - 8.41T + 59T^{2} \)
67 \( 1 - 0.158T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 - 7.62T + 79T^{2} \)
83 \( 1 - 8.04T + 83T^{2} \)
89 \( 1 + 6.83T + 89T^{2} \)
97 \( 1 + 2.44T + 97T^{2} \)
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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

−7.66175099856478678973779134676, −7.20311918247183627915128687599, −6.54456372234903449496812021570, −5.68002552912620180201939881610, −5.15886000139292652140130526617, −4.28285438106573131946146063226, −3.80341076913770943693434507620, −3.21278522029486421744923610800, −2.21568499666920742271401183581, −1.17089615182639809424965621164, 1.17089615182639809424965621164, 2.21568499666920742271401183581, 3.21278522029486421744923610800, 3.80341076913770943693434507620, 4.28285438106573131946146063226, 5.15886000139292652140130526617, 5.68002552912620180201939881610, 6.54456372234903449496812021570, 7.20311918247183627915128687599, 7.66175099856478678973779134676

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