Properties

Degree $2$
Conductor $4592$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 5-s − 7-s + 6·9-s + 2·11-s − 3·15-s − 3·17-s + 8·19-s − 3·21-s + 4·23-s − 4·25-s + 9·27-s − 5·29-s + 3·31-s + 6·33-s + 35-s + 10·37-s − 41-s + 5·43-s − 6·45-s − 6·47-s + 49-s − 9·51-s − 9·53-s − 2·55-s + 24·57-s + 10·59-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s + 0.603·11-s − 0.774·15-s − 0.727·17-s + 1.83·19-s − 0.654·21-s + 0.834·23-s − 4/5·25-s + 1.73·27-s − 0.928·29-s + 0.538·31-s + 1.04·33-s + 0.169·35-s + 1.64·37-s − 0.156·41-s + 0.762·43-s − 0.894·45-s − 0.875·47-s + 1/7·49-s − 1.26·51-s − 1.23·53-s − 0.269·55-s + 3.17·57-s + 1.30·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4592\)    =    \(2^{4} \cdot 7 \cdot 41\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4592} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4592,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.663527059\)
\(L(\frac12)\) \(\approx\) \(3.663527059\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
41 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−8.249185174589333893729289907121, −7.72997338592006774310578926588, −7.14560736165226729437317372106, −6.38622996785503235332944176293, −5.23622748209082737351087297488, −4.23478767674851637733024753016, −3.63453272138404773358328841667, −2.98993227622854412965642782912, −2.16050885343918737413216353123, −1.02861097940336389288142736785, 1.02861097940336389288142736785, 2.16050885343918737413216353123, 2.98993227622854412965642782912, 3.63453272138404773358328841667, 4.23478767674851637733024753016, 5.23622748209082737351087297488, 6.38622996785503235332944176293, 7.14560736165226729437317372106, 7.72997338592006774310578926588, 8.249185174589333893729289907121

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