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  + 3.24·3-s + 2.56·5-s + 0.858·7-s + 7.49·9-s − 6.20·11-s − 1.41·13-s + 8.31·15-s + 3.93·17-s − 3.82·19-s + 2.78·21-s + 3.06·23-s + 1.58·25-s + 14.5·27-s + 8.48·29-s + 1.13·31-s − 20.1·33-s + 2.20·35-s + 8.49·37-s − 4.58·39-s + 5.34·43-s + 19.2·45-s + 6.65·47-s − 6.26·49-s + 12.7·51-s − 6.41·53-s − 15.9·55-s − 12.3·57-s + ⋯
L(s)  = 1  + 1.87·3-s + 1.14·5-s + 0.324·7-s + 2.49·9-s − 1.87·11-s − 0.392·13-s + 2.14·15-s + 0.953·17-s − 0.877·19-s + 0.607·21-s + 0.639·23-s + 0.317·25-s + 2.80·27-s + 1.57·29-s + 0.203·31-s − 3.50·33-s + 0.372·35-s + 1.39·37-s − 0.734·39-s + 0.815·43-s + 2.86·45-s + 0.970·47-s − 0.894·49-s + 1.78·51-s − 0.881·53-s − 2.14·55-s − 1.64·57-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

\(L(1)\) \(\approx\) \(5.352079647\)
\(L(\frac12)\) \(\approx\) \(5.352079647\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.24T + 3T^{2} \)
5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 - 0.858T + 7T^{2} \)
11 \( 1 + 6.20T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 - 3.93T + 17T^{2} \)
19 \( 1 + 3.82T + 19T^{2} \)
23 \( 1 - 3.06T + 23T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 - 1.13T + 31T^{2} \)
37 \( 1 - 8.49T + 37T^{2} \)
43 \( 1 - 5.34T + 43T^{2} \)
47 \( 1 - 6.65T + 47T^{2} \)
53 \( 1 + 6.41T + 53T^{2} \)
59 \( 1 + 3.06T + 59T^{2} \)
61 \( 1 - 7.41T + 61T^{2} \)
67 \( 1 + 1.79T + 67T^{2} \)
71 \( 1 - 3.02T + 71T^{2} \)
73 \( 1 - 0.632T + 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + 8.76T + 83T^{2} \)
89 \( 1 - 7.89T + 89T^{2} \)
97 \( 1 + 9.86T + 97T^{2} \)
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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

−8.080300138351113416804570270532, −7.59727752208322979934892558779, −6.76010826476990944214778687170, −5.83768741327472535441693306632, −4.99474670608161952494673428685, −4.39027270593753717763490358459, −3.23238423637044424725660932578, −2.54778747143231244901315907242, −2.26734061779770868420806215508, −1.15205766768069528571859093686, 1.15205766768069528571859093686, 2.26734061779770868420806215508, 2.54778747143231244901315907242, 3.23238423637044424725660932578, 4.39027270593753717763490358459, 4.99474670608161952494673428685, 5.83768741327472535441693306632, 6.76010826476990944214778687170, 7.59727752208322979934892558779, 8.080300138351113416804570270532

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