Properties

Label 4-616e2-1.1-c1e2-0-47
Degree $4$
Conductor $379456$
Sign $-1$
Analytic cond. $24.1944$
Root an. cond. $2.21783$
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  + 4·5-s − 6·9-s − 4·11-s + 2·25-s + 16·31-s − 4·37-s − 24·45-s − 16·47-s + 49-s + 12·53-s − 16·55-s − 8·67-s − 16·71-s + 27·81-s − 12·89-s − 12·97-s + 24·99-s − 32·103-s + 4·113-s + 5·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.78·5-s − 2·9-s − 1.20·11-s + 2/5·25-s + 2.87·31-s − 0.657·37-s − 3.57·45-s − 2.33·47-s + 1/7·49-s + 1.64·53-s − 2.15·55-s − 0.977·67-s − 1.89·71-s + 3·81-s − 1.27·89-s − 1.21·97-s + 2.41·99-s − 3.15·103-s + 0.376·113-s + 5/11·121-s − 2.50·125-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(379456\)    =    \(2^{6} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(24.1944\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 379456,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−8.423245666401170068214022844193, −8.188420417684287628984093773695, −7.69587064441186150768078543731, −6.88310002274215763839569601337, −6.42262084813563967264199691637, −5.94974241082854154889853475745, −5.72315905059004231786170242312, −5.22316409435338364257345412913, −4.86884684581528043387154729760, −4.03151585237256402081332864489, −2.98780927444278559643957768594, −2.79183800612725741835263061431, −2.30074942792092931708066021417, −1.44245425010921288807179765849, 0, 1.44245425010921288807179765849, 2.30074942792092931708066021417, 2.79183800612725741835263061431, 2.98780927444278559643957768594, 4.03151585237256402081332864489, 4.86884684581528043387154729760, 5.22316409435338364257345412913, 5.72315905059004231786170242312, 5.94974241082854154889853475745, 6.42262084813563967264199691637, 6.88310002274215763839569601337, 7.69587064441186150768078543731, 8.188420417684287628984093773695, 8.423245666401170068214022844193

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