Properties

Degree $2$
Conductor $6036$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4.23·5-s − 0.513·7-s + 9-s − 2.27·11-s + 2.85·13-s − 4.23·15-s + 6.71·17-s + 6.76·19-s + 0.513·21-s + 0.132·23-s + 12.9·25-s − 27-s − 2.95·29-s − 2.49·31-s + 2.27·33-s − 2.17·35-s + 4.59·37-s − 2.85·39-s + 2.66·41-s + 8.46·43-s + 4.23·45-s − 5.38·47-s − 6.73·49-s − 6.71·51-s − 4.52·53-s − 9.64·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.89·5-s − 0.194·7-s + 0.333·9-s − 0.686·11-s + 0.792·13-s − 1.09·15-s + 1.62·17-s + 1.55·19-s + 0.112·21-s + 0.0276·23-s + 2.58·25-s − 0.192·27-s − 0.549·29-s − 0.447·31-s + 0.396·33-s − 0.367·35-s + 0.754·37-s − 0.457·39-s + 0.415·41-s + 1.29·43-s + 0.631·45-s − 0.784·47-s − 0.962·49-s − 0.940·51-s − 0.621·53-s − 1.30·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6036\)    =    \(2^{2} \cdot 3 \cdot 503\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6036} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6036,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.757687470\)
\(L(\frac12)\) \(\approx\) \(2.757687470\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
503 \( 1 + T \)
good5 \( 1 - 4.23T + 5T^{2} \)
7 \( 1 + 0.513T + 7T^{2} \)
11 \( 1 + 2.27T + 11T^{2} \)
13 \( 1 - 2.85T + 13T^{2} \)
17 \( 1 - 6.71T + 17T^{2} \)
19 \( 1 - 6.76T + 19T^{2} \)
23 \( 1 - 0.132T + 23T^{2} \)
29 \( 1 + 2.95T + 29T^{2} \)
31 \( 1 + 2.49T + 31T^{2} \)
37 \( 1 - 4.59T + 37T^{2} \)
41 \( 1 - 2.66T + 41T^{2} \)
43 \( 1 - 8.46T + 43T^{2} \)
47 \( 1 + 5.38T + 47T^{2} \)
53 \( 1 + 4.52T + 53T^{2} \)
59 \( 1 - 3.87T + 59T^{2} \)
61 \( 1 + 7.60T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + 7.80T + 71T^{2} \)
73 \( 1 - 7.62T + 73T^{2} \)
79 \( 1 - 7.12T + 79T^{2} \)
83 \( 1 - 4.86T + 83T^{2} \)
89 \( 1 + 3.93T + 89T^{2} \)
97 \( 1 - 5.48T + 97T^{2} \)
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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

−7.905739486182863084203328703369, −7.36812781788711675155468815134, −6.35584581152009347345860211127, −5.84466586445594037143140798351, −5.45702468034318644182195453489, −4.79267510346958939694965836838, −3.44372084199777880765204336094, −2.77556775280488823320375557760, −1.65687810650294570286912185829, −0.987414549523421277293288386132, 0.987414549523421277293288386132, 1.65687810650294570286912185829, 2.77556775280488823320375557760, 3.44372084199777880765204336094, 4.79267510346958939694965836838, 5.45702468034318644182195453489, 5.84466586445594037143140798351, 6.35584581152009347345860211127, 7.36812781788711675155468815134, 7.905739486182863084203328703369

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