Properties

Label 4-112e3-1.1-c1e2-0-20
Degree $4$
Conductor $1404928$
Sign $-1$
Analytic cond. $89.5794$
Root an. cond. $3.07646$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 2·9-s − 10·25-s + 12·29-s − 4·37-s + 16·43-s + 49-s − 12·53-s + 2·63-s − 8·67-s − 16·79-s − 5·81-s + 24·107-s − 4·109-s + 12·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + ⋯
L(s)  = 1  − 0.377·7-s − 2/3·9-s − 2·25-s + 2.22·29-s − 0.657·37-s + 2.43·43-s + 1/7·49-s − 1.64·53-s + 0.251·63-s − 0.977·67-s − 1.80·79-s − 5/9·81-s + 2.32·107-s − 0.383·109-s + 1.12·113-s − 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-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: no
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.68957389720506083723150978195, −7.29200576171944643329126204500, −6.96175265373852036821224883683, −6.17368924384109906924363362759, −6.04992362572003306544038558924, −5.73625972355331222443538857924, −5.06701824847582369879735343431, −4.39126934619586232082133319537, −4.31547623044891473234039816601, −3.42443184461883656887541100921, −3.09628357761679869111062402047, −2.49739457868022289115900838847, −1.90996831082337002056034743405, −1.00403210582709168291350506092, 0, 1.00403210582709168291350506092, 1.90996831082337002056034743405, 2.49739457868022289115900838847, 3.09628357761679869111062402047, 3.42443184461883656887541100921, 4.31547623044891473234039816601, 4.39126934619586232082133319537, 5.06701824847582369879735343431, 5.73625972355331222443538857924, 6.04992362572003306544038558924, 6.17368924384109906924363362759, 6.96175265373852036821224883683, 7.29200576171944643329126204500, 7.68957389720506083723150978195

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