Properties

Label 4-96e4-1.1-c1e2-0-14
Degree $4$
Conductor $84934656$
Sign $1$
Analytic cond. $5415.50$
Root an. cond. $8.57846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s + 8·23-s − 8·25-s + 16·31-s − 16·41-s + 16·47-s + 34·49-s + 24·71-s − 4·73-s − 28·89-s − 32·97-s + 8·103-s + 4·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 64·161-s + 163-s + 167-s − 8·169-s + 173-s + 64·175-s + ⋯
L(s)  = 1  − 3.02·7-s + 1.66·23-s − 8/5·25-s + 2.87·31-s − 2.49·41-s + 2.33·47-s + 34/7·49-s + 2.84·71-s − 0.468·73-s − 2.96·89-s − 3.24·97-s + 0.788·103-s + 0.376·113-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 + 0.0798·157-s − 5.04·161-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + 0.0760·173-s + 4.83·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84934656\)    =    \(2^{20} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(5415.50\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 84934656,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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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

−7.27780913891541929531000437517, −7.07301984858546627841061214149, −6.76504835716971797125268641670, −6.70055957396911924560523386720, −6.15497763254524788789051150949, −6.02267485692751963948975366251, −5.44041990646450752544189048056, −5.42096989634802736110553616880, −4.54492723569702910322371839102, −4.51460878845409033536914591396, −3.82986914363528076864579788231, −3.54120420949680834573635082079, −3.31431179032126352091070176315, −2.88468196564714809973376165956, −2.40475789334497589012328804368, −2.36549804664025002576758092015, −1.18403239672901636861719729853, −1.01033670075853047259470332273, 0, 0, 1.01033670075853047259470332273, 1.18403239672901636861719729853, 2.36549804664025002576758092015, 2.40475789334497589012328804368, 2.88468196564714809973376165956, 3.31431179032126352091070176315, 3.54120420949680834573635082079, 3.82986914363528076864579788231, 4.51460878845409033536914591396, 4.54492723569702910322371839102, 5.42096989634802736110553616880, 5.44041990646450752544189048056, 6.02267485692751963948975366251, 6.15497763254524788789051150949, 6.70055957396911924560523386720, 6.76504835716971797125268641670, 7.07301984858546627841061214149, 7.27780913891541929531000437517

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