Properties

Label 4-96e4-1.1-c1e2-0-21
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

−7.53413814462915810572819498920, −7.45614250003428663403538213435, −6.83462245430182591544481842943, −6.75011339841494960825337771469, −5.94791332206254764964182323358, −5.75968663101382573921778867059, −5.34910991200909398195537998534, −5.25678139234700986992934751580, −4.82821336191601818257543150790, −4.33692588222372219242358122420, −4.03975531900275793108932488479, −3.94949651072399384362782792622, −3.23535578129751299135920327957, −2.84159011125453390325761343219, −2.08172054214496432443369192771, −1.77467576648189249407772828247, −1.54489307620531675471409139468, −1.48823445499472879462712629739, 0, 0, 1.48823445499472879462712629739, 1.54489307620531675471409139468, 1.77467576648189249407772828247, 2.08172054214496432443369192771, 2.84159011125453390325761343219, 3.23535578129751299135920327957, 3.94949651072399384362782792622, 4.03975531900275793108932488479, 4.33692588222372219242358122420, 4.82821336191601818257543150790, 5.25678139234700986992934751580, 5.34910991200909398195537998534, 5.75968663101382573921778867059, 5.94791332206254764964182323358, 6.75011339841494960825337771469, 6.83462245430182591544481842943, 7.45614250003428663403538213435, 7.53413814462915810572819498920

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