Properties

Label 4-48e4-1.1-c1e2-0-26
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $338.469$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

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: $1$
Analytic conductor: \(338.469\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.031968381\)
\(L(\frac12)\) \(\approx\) \(5.031968381\)
\(L(\frac{3}{2})\) 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

−8.793760956749197758015793851178, −8.782890792057012148717923048519, −8.509755686003725234299752697398, −8.117736686076816284090835377516, −7.64161367339961920027084258393, −7.29963258870782483710472540334, −7.12309300166512313565346938203, −6.62180298459201982165834510160, −5.84930303477066848876127724748, −5.48692317258635396260497518279, −5.23660894146425618050434861345, −4.85617985742319696959468116782, −4.41324315214423738789174059540, −4.24199522480338224179017852798, −3.32705409726734369332605831933, −3.03417653043187040154959811955, −2.28141504243485475984353723068, −1.81598920577219138278859064481, −1.22108427632985252910194514225, −0.923414068528069850584061886816, 0.923414068528069850584061886816, 1.22108427632985252910194514225, 1.81598920577219138278859064481, 2.28141504243485475984353723068, 3.03417653043187040154959811955, 3.32705409726734369332605831933, 4.24199522480338224179017852798, 4.41324315214423738789174059540, 4.85617985742319696959468116782, 5.23660894146425618050434861345, 5.48692317258635396260497518279, 5.84930303477066848876127724748, 6.62180298459201982165834510160, 7.12309300166512313565346938203, 7.29963258870782483710472540334, 7.64161367339961920027084258393, 8.117736686076816284090835377516, 8.509755686003725234299752697398, 8.782890792057012148717923048519, 8.793760956749197758015793851178

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