Properties

Label 4-1216e2-1.1-c1e2-0-18
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
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  + 8·5-s − 5·9-s + 2·13-s + 6·17-s + 38·25-s + 10·29-s + 4·37-s − 16·41-s − 40·45-s − 5·49-s + 2·53-s − 4·61-s + 16·65-s + 18·73-s + 16·81-s + 48·85-s − 4·97-s − 4·101-s + 30·109-s + 28·113-s − 10·117-s − 18·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 3.57·5-s − 5/3·9-s + 0.554·13-s + 1.45·17-s + 38/5·25-s + 1.85·29-s + 0.657·37-s − 2.49·41-s − 5.96·45-s − 5/7·49-s + 0.274·53-s − 0.512·61-s + 1.98·65-s + 2.10·73-s + 16/9·81-s + 5.20·85-s − 0.406·97-s − 0.398·101-s + 2.87·109-s + 2.63·113-s − 0.924·117-s − 1.63·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1478656} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.825279813\)
\(L(\frac12)\) \(\approx\) \(4.825279813\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−8.264845552131759886230905508728, −7.41068992206199799917476516665, −6.67486851525401293685577568233, −6.41602681477419049736409749590, −6.17646264461499634926610391599, −5.70311644291868843897751049539, −5.50702009116960948045704530045, −4.89838836690724431377597203004, −4.86338510531971803230901631456, −3.35219753314436267860210658260, −3.27846026315236304587310905813, −2.45747388367035991238545338827, −2.31224961625843070084602752549, −1.49207601742473373563863301755, −1.03106032747078173831338248002, 1.03106032747078173831338248002, 1.49207601742473373563863301755, 2.31224961625843070084602752549, 2.45747388367035991238545338827, 3.27846026315236304587310905813, 3.35219753314436267860210658260, 4.86338510531971803230901631456, 4.89838836690724431377597203004, 5.50702009116960948045704530045, 5.70311644291868843897751049539, 6.17646264461499634926610391599, 6.41602681477419049736409749590, 6.67486851525401293685577568233, 7.41068992206199799917476516665, 8.264845552131759886230905508728

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