Properties

Label 4-96e4-1.1-c1e2-0-20
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  + 4·7-s − 4·17-s − 12·23-s − 8·25-s + 16·31-s − 16·47-s − 2·49-s − 20·71-s + 8·73-s − 8·89-s − 4·97-s − 12·103-s − 12·113-s − 16·119-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s − 24·169-s + 173-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.970·17-s − 2.50·23-s − 8/5·25-s + 2.87·31-s − 2.33·47-s − 2/7·49-s − 2.37·71-s + 0.936·73-s − 0.847·89-s − 0.406·97-s − 1.18·103-s − 1.12·113-s − 1.46·119-s − 1.81·121-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 − 3.78·161-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 84 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 164 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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

−7.51492538936526841986057630477, −7.48654113349428805147724410845, −6.73082787597525248820353617691, −6.45742913161854301633481661620, −6.16449417391641614736057879116, −6.03865837456829704952567991427, −5.26907252264224647150774383578, −5.24904918684148510310328686897, −4.58477741635161640209811755153, −4.52351514702833609690323614937, −4.05969940102495599980727480232, −3.87882403593986685800817551240, −3.19423179007541018340507254649, −2.78097724796515450927844404860, −2.21747932060423342208001006657, −2.05668401343670540566030108067, −1.38943510269200836088248952408, −1.29403526790407359603601570518, 0, 0, 1.29403526790407359603601570518, 1.38943510269200836088248952408, 2.05668401343670540566030108067, 2.21747932060423342208001006657, 2.78097724796515450927844404860, 3.19423179007541018340507254649, 3.87882403593986685800817551240, 4.05969940102495599980727480232, 4.52351514702833609690323614937, 4.58477741635161640209811755153, 5.24904918684148510310328686897, 5.26907252264224647150774383578, 6.03865837456829704952567991427, 6.16449417391641614736057879116, 6.45742913161854301633481661620, 6.73082787597525248820353617691, 7.48654113349428805147724410845, 7.51492538936526841986057630477

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