Properties

Label 4-912e2-1.1-c1e2-0-97
Degree $4$
Conductor $831744$
Sign $1$
Analytic cond. $53.0327$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 6·5-s − 2·7-s + 6·9-s + 9·13-s + 18·15-s − 6·17-s + 8·19-s − 6·21-s + 19·25-s + 9·27-s − 6·29-s − 12·35-s + 27·39-s − 12·41-s − 43-s + 36·45-s − 12·47-s − 11·49-s − 18·51-s − 12·53-s + 24·57-s − 7·61-s − 12·63-s + 54·65-s − 15·67-s + 6·71-s + ⋯
L(s)  = 1  + 1.73·3-s + 2.68·5-s − 0.755·7-s + 2·9-s + 2.49·13-s + 4.64·15-s − 1.45·17-s + 1.83·19-s − 1.30·21-s + 19/5·25-s + 1.73·27-s − 1.11·29-s − 2.02·35-s + 4.32·39-s − 1.87·41-s − 0.152·43-s + 5.36·45-s − 1.75·47-s − 1.57·49-s − 2.52·51-s − 1.64·53-s + 3.17·57-s − 0.896·61-s − 1.51·63-s + 6.69·65-s − 1.83·67-s + 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(831744\)    =    \(2^{8} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(53.0327\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 831744,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.092960273\)
\(L(\frac12)\) \(\approx\) \(7.092960273\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 95 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−9.928233454302836245203350097863, −9.669219937949775089496725376886, −9.595050222532212835509216804470, −9.095254492352062399537752274407, −8.670108693975763968142596658673, −8.584785539635644445243430999637, −7.88193314646383327811318753797, −7.37647689978811065529997490085, −6.61508926834439222043164833043, −6.55343454635500180073030035466, −6.00499403050244054769062561264, −5.78190990133850197054414974308, −4.95657816502970387590339440363, −4.63733297448008114901755583472, −3.51084002464723050754903134445, −3.37667335065790969181971061924, −3.01652075159725567265311669795, −2.11295748302503760662776943399, −1.65629867996553385547471978989, −1.44884665951825694270658039340, 1.44884665951825694270658039340, 1.65629867996553385547471978989, 2.11295748302503760662776943399, 3.01652075159725567265311669795, 3.37667335065790969181971061924, 3.51084002464723050754903134445, 4.63733297448008114901755583472, 4.95657816502970387590339440363, 5.78190990133850197054414974308, 6.00499403050244054769062561264, 6.55343454635500180073030035466, 6.61508926834439222043164833043, 7.37647689978811065529997490085, 7.88193314646383327811318753797, 8.584785539635644445243430999637, 8.670108693975763968142596658673, 9.095254492352062399537752274407, 9.595050222532212835509216804470, 9.669219937949775089496725376886, 9.928233454302836245203350097863

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