Properties

Degree $4$
Conductor $5308416$
Sign $1$
Motivic weight $1$
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 + 4·17-s − 8·23-s + 6·25-s − 12·41-s − 16·47-s + 34·49-s − 24·71-s − 28·73-s + 16·79-s − 4·89-s − 4·97-s − 8·103-s − 4·113-s − 32·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 64·161-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  − 3.02·7-s + 0.970·17-s − 1.66·23-s + 6/5·25-s − 1.87·41-s − 2.33·47-s + 34/7·49-s − 2.84·71-s − 3.27·73-s + 1.80·79-s − 0.423·89-s − 0.406·97-s − 0.788·103-s − 0.376·113-s − 2.93·119-s + 1.63·121-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 + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5308416,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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

−8.901283822911148933219274961961, −8.537883598187149394139660939031, −7.927597082960768217010357813251, −7.78807401181537754550940713269, −6.90658693472445454577262700351, −6.88483364047604802017468384205, −6.63217600020591710133260327071, −6.05413092799802394714867394531, −5.75245531885267310765400026046, −5.54545642580210773785881917936, −4.65041292432824869473491905227, −4.43840339046971541107811169446, −3.54504386175201089083978945511, −3.51512988593556574794619540593, −3.02134894909850940868110394782, −2.76949698107409344784867387252, −1.89489278127510207413757070938, −1.21146561810893222769175172833, 0, 0, 1.21146561810893222769175172833, 1.89489278127510207413757070938, 2.76949698107409344784867387252, 3.02134894909850940868110394782, 3.51512988593556574794619540593, 3.54504386175201089083978945511, 4.43840339046971541107811169446, 4.65041292432824869473491905227, 5.54545642580210773785881917936, 5.75245531885267310765400026046, 6.05413092799802394714867394531, 6.63217600020591710133260327071, 6.88483364047604802017468384205, 6.90658693472445454577262700351, 7.78807401181537754550940713269, 7.927597082960768217010357813251, 8.537883598187149394139660939031, 8.901283822911148933219274961961

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