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  − 2.21·3-s + 3.59·5-s + 5.06·7-s + 1.90·9-s + 2.55·11-s + 4.93·13-s − 7.96·15-s − 2.68·17-s − 4.72·19-s − 11.2·21-s − 1.49·23-s + 7.93·25-s + 2.41·27-s − 2.43·29-s + 3.19·31-s − 5.67·33-s + 18.2·35-s + 2.90·37-s − 10.9·39-s − 7.62·43-s + 6.86·45-s − 5.15·47-s + 18.6·49-s + 5.95·51-s + 11.1·53-s + 9.20·55-s + 10.4·57-s + ⋯
L(s)  = 1  − 1.27·3-s + 1.60·5-s + 1.91·7-s + 0.636·9-s + 0.771·11-s + 1.36·13-s − 2.05·15-s − 0.651·17-s − 1.08·19-s − 2.44·21-s − 0.311·23-s + 1.58·25-s + 0.465·27-s − 0.451·29-s + 0.573·31-s − 0.987·33-s + 3.07·35-s + 0.478·37-s − 1.75·39-s − 1.16·43-s + 1.02·45-s − 0.751·47-s + 2.66·49-s + 0.833·51-s + 1.52·53-s + 1.24·55-s + 1.38·57-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

\(L(1)\) \(\approx\) \(2.604019520\)
\(L(\frac12)\) \(\approx\) \(2.604019520\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.21T + 3T^{2} \)
5 \( 1 - 3.59T + 5T^{2} \)
7 \( 1 - 5.06T + 7T^{2} \)
11 \( 1 - 2.55T + 11T^{2} \)
13 \( 1 - 4.93T + 13T^{2} \)
17 \( 1 + 2.68T + 17T^{2} \)
19 \( 1 + 4.72T + 19T^{2} \)
23 \( 1 + 1.49T + 23T^{2} \)
29 \( 1 + 2.43T + 29T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 - 2.90T + 37T^{2} \)
43 \( 1 + 7.62T + 43T^{2} \)
47 \( 1 + 5.15T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 1.49T + 59T^{2} \)
61 \( 1 - 1.06T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 8.28T + 73T^{2} \)
79 \( 1 - 4.28T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + 9.36T + 89T^{2} \)
97 \( 1 - 3.37T + 97T^{2} \)
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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

−8.193336753382257660749269551140, −6.93538673515830858185224155073, −6.32686793550529663790725564068, −5.92921898557324499568839671137, −5.21971392893576506600918901737, −4.68537200409089096738632466319, −3.89458075633052435173962312151, −2.29014701559842378351435942418, −1.66649080093027739177911646094, −0.983130730068416537128204228536, 0.983130730068416537128204228536, 1.66649080093027739177911646094, 2.29014701559842378351435942418, 3.89458075633052435173962312151, 4.68537200409089096738632466319, 5.21971392893576506600918901737, 5.92921898557324499568839671137, 6.32686793550529663790725564068, 6.93538673515830858185224155073, 8.193336753382257660749269551140

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