Properties

Degree $2$
Conductor $6724$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0950·3-s + 1.17·5-s − 3.14·7-s − 2.99·9-s + 1.67·11-s − 6.63·13-s − 0.111·15-s − 5.16·17-s + 4.72·19-s + 0.299·21-s − 8.82·23-s − 3.63·25-s + 0.569·27-s + 1.80·29-s − 1.65·31-s − 0.159·33-s − 3.68·35-s − 1.99·37-s + 0.630·39-s + 1.46·43-s − 3.50·45-s + 8.53·47-s + 2.89·49-s + 0.490·51-s + 9.35·53-s + 1.96·55-s − 0.449·57-s + ⋯
L(s)  = 1  − 0.0549·3-s + 0.523·5-s − 1.18·7-s − 0.996·9-s + 0.505·11-s − 1.83·13-s − 0.0287·15-s − 1.25·17-s + 1.08·19-s + 0.0652·21-s − 1.83·23-s − 0.726·25-s + 0.109·27-s + 0.336·29-s − 0.298·31-s − 0.0277·33-s − 0.622·35-s − 0.327·37-s + 0.100·39-s + 0.224·43-s − 0.521·45-s + 1.24·47-s + 0.413·49-s + 0.0687·51-s + 1.28·53-s + 0.264·55-s − 0.0595·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6724\)    =    \(2^{2} \cdot 41^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6724} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6724,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8081021128\)
\(L(\frac12)\) \(\approx\) \(0.8081021128\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + 0.0950T + 3T^{2} \)
5 \( 1 - 1.17T + 5T^{2} \)
7 \( 1 + 3.14T + 7T^{2} \)
11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 + 6.63T + 13T^{2} \)
17 \( 1 + 5.16T + 17T^{2} \)
19 \( 1 - 4.72T + 19T^{2} \)
23 \( 1 + 8.82T + 23T^{2} \)
29 \( 1 - 1.80T + 29T^{2} \)
31 \( 1 + 1.65T + 31T^{2} \)
37 \( 1 + 1.99T + 37T^{2} \)
43 \( 1 - 1.46T + 43T^{2} \)
47 \( 1 - 8.53T + 47T^{2} \)
53 \( 1 - 9.35T + 53T^{2} \)
59 \( 1 - 8.82T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 + 9.67T + 67T^{2} \)
71 \( 1 - 0.776T + 71T^{2} \)
73 \( 1 - 8.33T + 73T^{2} \)
79 \( 1 + 0.915T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 6.44T + 89T^{2} \)
97 \( 1 - 8.32T + 97T^{2} \)
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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

−7.934401483132412481268264096329, −7.14418536636324326154862830728, −6.59223463536567917801343095959, −5.81202799901677398944911707310, −5.38017910054005335566593201670, −4.32628592666445124257855700736, −3.55904857843142462112418169405, −2.55265154960699881896491146697, −2.15201121867836451447185762515, −0.42628324180809372064517322649, 0.42628324180809372064517322649, 2.15201121867836451447185762515, 2.55265154960699881896491146697, 3.55904857843142462112418169405, 4.32628592666445124257855700736, 5.38017910054005335566593201670, 5.81202799901677398944911707310, 6.59223463536567917801343095959, 7.14418536636324326154862830728, 7.934401483132412481268264096329

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