Properties

Label 4-1142784-1.1-c1e2-0-8
Degree $4$
Conductor $1142784$
Sign $-1$
Analytic cond. $72.8648$
Root an. cond. $2.92165$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s − 2·9-s + 3·13-s − 19-s + 21-s + 25-s − 5·27-s − 4·31-s − 9·37-s + 3·39-s + 2·43-s − 11·49-s − 57-s + 9·61-s − 2·63-s − 6·67-s − 11·73-s + 75-s + 79-s + 81-s + 3·91-s − 4·93-s − 6·97-s + 21·103-s + 7·109-s − 9·111-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.832·13-s − 0.229·19-s + 0.218·21-s + 1/5·25-s − 0.962·27-s − 0.718·31-s − 1.47·37-s + 0.480·39-s + 0.304·43-s − 1.57·49-s − 0.132·57-s + 1.15·61-s − 0.251·63-s − 0.733·67-s − 1.28·73-s + 0.115·75-s + 0.112·79-s + 1/9·81-s + 0.314·91-s − 0.414·93-s − 0.609·97-s + 2.06·103-s + 0.670·109-s − 0.854·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1142784\)    =    \(2^{12} \cdot 3^{2} \cdot 31\)
Sign: $-1$
Analytic conductor: \(72.8648\)
Root analytic conductor: \(2.92165\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1142784} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1142784,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 5 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.932364995635833004904834931729, −7.42830552079449870980719386313, −7.09351010203907541383746147181, −6.38168043282382556604658362115, −6.21327324192910162941231483791, −5.49113311877190080836796921283, −5.27259666667352935116883900228, −4.63692943221780377838055091210, −4.08552887414332821773601851516, −3.48465830993130161851242594785, −3.24084459906268147169457614140, −2.48932756091283111397588211070, −1.91097765562853954472273704087, −1.26489281741664755971311219616, 0, 1.26489281741664755971311219616, 1.91097765562853954472273704087, 2.48932756091283111397588211070, 3.24084459906268147169457614140, 3.48465830993130161851242594785, 4.08552887414332821773601851516, 4.63692943221780377838055091210, 5.27259666667352935116883900228, 5.49113311877190080836796921283, 6.21327324192910162941231483791, 6.38168043282382556604658362115, 7.09351010203907541383746147181, 7.42830552079449870980719386313, 7.932364995635833004904834931729

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