Properties

Degree $4$
Conductor $20736$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 3·7-s − 3·9-s + 3·11-s + 5·13-s + 9·23-s + 25-s + 3·29-s − 9·31-s − 9·35-s + 4·37-s − 9·41-s + 9·43-s − 9·45-s + 3·47-s − 49-s + 9·55-s − 3·59-s + 61-s + 9·63-s + 15·65-s + 15·67-s − 24·71-s − 4·73-s − 9·77-s − 15·79-s + 9·81-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.13·7-s − 9-s + 0.904·11-s + 1.38·13-s + 1.87·23-s + 1/5·25-s + 0.557·29-s − 1.61·31-s − 1.52·35-s + 0.657·37-s − 1.40·41-s + 1.37·43-s − 1.34·45-s + 0.437·47-s − 1/7·49-s + 1.21·55-s − 0.390·59-s + 0.128·61-s + 1.13·63-s + 1.86·65-s + 1.83·67-s − 2.84·71-s − 0.468·73-s − 1.02·77-s − 1.68·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20736\)    =    \(2^{8} \cdot 3^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{144} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 20736,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32209\)
\(L(\frac12)\) \(\approx\) \(1.32209\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.24115208490608309117435303920, −13.22180926602360114803102296989, −12.40152109552675379619288166045, −11.87905048699547129785020985695, −11.20291703544438106782868112281, −10.84736513457373417686869429970, −10.34028644547091426521377671830, −9.511677227347317335941811743246, −9.312944172572840639402570621401, −8.885870633594408685217822171037, −8.361474323775439297448556447842, −7.39513206385297268528046215994, −6.53807913591060777441140676272, −6.46462364300244324935226542980, −5.65049118824998910766678737467, −5.39849902506889789807652657025, −4.14480418184802737496489738642, −3.35383496393844425103258497966, −2.71520314460127089964457787895, −1.43834980079011542247287906633, 1.43834980079011542247287906633, 2.71520314460127089964457787895, 3.35383496393844425103258497966, 4.14480418184802737496489738642, 5.39849902506889789807652657025, 5.65049118824998910766678737467, 6.46462364300244324935226542980, 6.53807913591060777441140676272, 7.39513206385297268528046215994, 8.361474323775439297448556447842, 8.885870633594408685217822171037, 9.312944172572840639402570621401, 9.511677227347317335941811743246, 10.34028644547091426521377671830, 10.84736513457373417686869429970, 11.20291703544438106782868112281, 11.87905048699547129785020985695, 12.40152109552675379619288166045, 13.22180926602360114803102296989, 13.24115208490608309117435303920

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