Properties

Label 4-112e3-1.1-c1e2-0-26
Degree $4$
Conductor $1404928$
Sign $-1$
Analytic cond. $89.5794$
Root an. cond. $3.07646$
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  − 7-s − 4·9-s + 6·11-s − 2·23-s − 2·25-s + 4·29-s + 8·37-s − 10·43-s + 49-s − 2·53-s + 4·63-s + 8·67-s − 16·71-s − 6·77-s − 4·79-s + 7·81-s − 24·99-s − 12·107-s − 20·109-s − 16·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.377·7-s − 4/3·9-s + 1.80·11-s − 0.417·23-s − 2/5·25-s + 0.742·29-s + 1.31·37-s − 1.52·43-s + 1/7·49-s − 0.274·53-s + 0.503·63-s + 0.977·67-s − 1.89·71-s − 0.683·77-s − 0.450·79-s + 7/9·81-s − 2.41·99-s − 1.16·107-s − 1.91·109-s − 1.50·113-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1404928\)    =    \(2^{12} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(89.5794\)
Root analytic conductor: \(3.07646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1404928,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_1$ \( 1 + T \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
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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

−7.906062403938228542679312398505, −7.21266210692081217760995062289, −6.72266199817642079436113278419, −6.34042616063462477383315538038, −6.19301546835800892347944132928, −5.52097689825881924367323162295, −5.22610854033153588338685358423, −4.41018481923210917656030901512, −4.12314550792245600494335570182, −3.57963294693398734387319060901, −3.01206551410525039614809972986, −2.60422443330861622954426319001, −1.76501277401255988744889364793, −1.09057764479192131940372108563, 0, 1.09057764479192131940372108563, 1.76501277401255988744889364793, 2.60422443330861622954426319001, 3.01206551410525039614809972986, 3.57963294693398734387319060901, 4.12314550792245600494335570182, 4.41018481923210917656030901512, 5.22610854033153588338685358423, 5.52097689825881924367323162295, 6.19301546835800892347944132928, 6.34042616063462477383315538038, 6.72266199817642079436113278419, 7.21266210692081217760995062289, 7.906062403938228542679312398505

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