Properties

Label 4-1476e2-1.1-c1e2-0-1
Degree $4$
Conductor $2178576$
Sign $1$
Analytic cond. $138.907$
Root an. cond. $3.43306$
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  − 8·5-s − 8·23-s + 38·25-s − 6·31-s + 2·37-s + 8·41-s − 18·43-s − 2·49-s − 16·59-s + 14·61-s − 26·73-s − 24·83-s + 22·103-s − 8·113-s + 64·115-s + 13·121-s − 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·155-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3.57·5-s − 1.66·23-s + 38/5·25-s − 1.07·31-s + 0.328·37-s + 1.24·41-s − 2.74·43-s − 2/7·49-s − 2.08·59-s + 1.79·61-s − 3.04·73-s − 2.63·83-s + 2.16·103-s − 0.752·113-s + 5.96·115-s + 1.18·121-s − 12.1·125-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 + 3.85·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2178576\)    =    \(2^{4} \cdot 3^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(138.907\)
Root analytic conductor: \(3.43306\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2178576,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
41$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 141 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \)
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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

−9.102057242194324532889608306317, −8.583186023153649165460373595080, −8.449094410328137953880527944836, −8.118777053742042285317924728597, −7.59073884818765709926116770914, −7.53160673080902664424401454310, −6.98490287126549412715468122986, −6.79398682899539228030846400398, −5.95394242748069983084251539736, −5.64823513073751212840566002703, −4.73907683203477975564294956292, −4.49003882994578320873290918364, −4.28259567976784924609768445875, −3.66761260215031205692779651330, −3.33024698704583651930601066548, −3.06824639117645043782410067366, −2.07191517393699550808111049291, −1.13348383262806300238633910171, 0, 0, 1.13348383262806300238633910171, 2.07191517393699550808111049291, 3.06824639117645043782410067366, 3.33024698704583651930601066548, 3.66761260215031205692779651330, 4.28259567976784924609768445875, 4.49003882994578320873290918364, 4.73907683203477975564294956292, 5.64823513073751212840566002703, 5.95394242748069983084251539736, 6.79398682899539228030846400398, 6.98490287126549412715468122986, 7.53160673080902664424401454310, 7.59073884818765709926116770914, 8.118777053742042285317924728597, 8.449094410328137953880527944836, 8.583186023153649165460373595080, 9.102057242194324532889608306317

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