Properties

Label 4-1142784-1.1-c1e2-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s − 2·9-s + 7·13-s − 5·19-s + 21-s + 25-s − 5·27-s + 8·31-s − 37-s + 7·39-s + 2·43-s − 7·49-s − 5·57-s + 61-s − 2·63-s + 14·67-s − 7·73-s + 75-s + 5·79-s + 81-s + 7·91-s + 8·93-s + 6·97-s − 7·103-s − 5·109-s − 111-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 2/3·9-s + 1.94·13-s − 1.14·19-s + 0.218·21-s + 1/5·25-s − 0.962·27-s + 1.43·31-s − 0.164·37-s + 1.12·39-s + 0.304·43-s − 49-s − 0.662·57-s + 0.128·61-s − 0.251·63-s + 1.71·67-s − 0.819·73-s + 0.115·75-s + 0.562·79-s + 1/9·81-s + 0.733·91-s + 0.829·93-s + 0.609·97-s − 0.689·103-s − 0.478·109-s − 0.0949·111-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: \(0\)
Selberg data: \((4,\ 1142784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.658927385\)
\(L(\frac12)\) \(\approx\) \(2.658927385\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 - 7 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 72 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.099019925669871420899449770436, −7.951212596030188259642977683744, −7.15814873220015976581596522215, −6.64824623564342855796275616133, −6.31010011945207620075425791788, −5.92629860007276033402748165187, −5.45363719479375583298028614978, −4.85785522232237836938717516607, −4.29938770688531445955014253560, −3.89112949214453868107542740841, −3.33145212151059683709197545430, −2.88422434592188062474744063642, −2.18571663082976232428626251441, −1.60889689728010078756885159562, −0.72866282716440172154646166569, 0.72866282716440172154646166569, 1.60889689728010078756885159562, 2.18571663082976232428626251441, 2.88422434592188062474744063642, 3.33145212151059683709197545430, 3.89112949214453868107542740841, 4.29938770688531445955014253560, 4.85785522232237836938717516607, 5.45363719479375583298028614978, 5.92629860007276033402748165187, 6.31010011945207620075425791788, 6.64824623564342855796275616133, 7.15814873220015976581596522215, 7.951212596030188259642977683744, 8.099019925669871420899449770436

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