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 − 2.57·5-s − 0.722·7-s + 9-s − 1.16·11-s + 2.87·13-s + 2.57·15-s + 6.88·17-s + 0.0742·19-s + 0.722·21-s + 6.64·23-s + 1.61·25-s − 27-s − 8.36·29-s − 6.34·31-s + 1.16·33-s + 1.85·35-s − 4.23·37-s − 2.87·39-s + 1.08·41-s + 3.62·43-s − 2.57·45-s − 1.46·47-s − 6.47·49-s − 6.88·51-s + 9.12·53-s + 2.99·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.15·5-s − 0.273·7-s + 0.333·9-s − 0.350·11-s + 0.798·13-s + 0.664·15-s + 1.66·17-s + 0.0170·19-s + 0.157·21-s + 1.38·23-s + 0.323·25-s − 0.192·27-s − 1.55·29-s − 1.13·31-s + 0.202·33-s + 0.314·35-s − 0.696·37-s − 0.461·39-s + 0.168·41-s + 0.552·43-s − 0.383·45-s − 0.213·47-s − 0.925·49-s − 0.963·51-s + 1.25·53-s + 0.403·55-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

\(L(1)\) \(\approx\) \(1.013528716\)
\(L(\frac12)\) \(\approx\) \(1.013528716\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.57T + 5T^{2} \)
7 \( 1 + 0.722T + 7T^{2} \)
11 \( 1 + 1.16T + 11T^{2} \)
13 \( 1 - 2.87T + 13T^{2} \)
17 \( 1 - 6.88T + 17T^{2} \)
19 \( 1 - 0.0742T + 19T^{2} \)
23 \( 1 - 6.64T + 23T^{2} \)
29 \( 1 + 8.36T + 29T^{2} \)
31 \( 1 + 6.34T + 31T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 - 1.08T + 41T^{2} \)
43 \( 1 - 3.62T + 43T^{2} \)
47 \( 1 + 1.46T + 47T^{2} \)
53 \( 1 - 9.12T + 53T^{2} \)
59 \( 1 + 4.35T + 59T^{2} \)
61 \( 1 - 5.87T + 61T^{2} \)
67 \( 1 - 0.793T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 9.16T + 73T^{2} \)
79 \( 1 + 6.61T + 79T^{2} \)
83 \( 1 + 9.56T + 83T^{2} \)
89 \( 1 + 9.56T + 89T^{2} \)
97 \( 1 - 1.48T + 97T^{2} \)
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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

−7.903766416111653577025615697009, −7.37895156258811623326060268295, −6.83644388677576789324201100988, −5.67147600003572098022877945889, −5.45157492344764787567385707875, −4.35831287118463714476223427231, −3.60819121444589188735497216284, −3.11809587177218490964869130429, −1.61380551175231004243264897047, −0.56266378842928157673989980928, 0.56266378842928157673989980928, 1.61380551175231004243264897047, 3.11809587177218490964869130429, 3.60819121444589188735497216284, 4.35831287118463714476223427231, 5.45157492344764787567385707875, 5.67147600003572098022877945889, 6.83644388677576789324201100988, 7.37895156258811623326060268295, 7.903766416111653577025615697009

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