Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{6} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 4·19-s + 7·25-s − 12·29-s − 20·31-s − 22·37-s − 18·47-s + 9·49-s + 24·53-s − 6·59-s − 30·83-s − 8·103-s + 10·109-s + 36·113-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.917·19-s + 7/5·25-s − 2.22·29-s − 3.59·31-s − 3.61·37-s − 2.62·47-s + 9/7·49-s + 3.29·53-s − 0.781·59-s − 3.29·83-s − 0.788·103-s + 0.957·109-s + 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.245626230\)
\(L(\frac12)\)  \(\approx\)  \(1.245626230\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 79 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.811972540626731772952779411353, −8.647074114512688486035137679657, −8.330648532192857778115414626618, −7.73036194167570246420712176758, −7.19035527388077362313746820214, −7.08284676861751351858076882137, −7.07659731465896497768463635599, −6.18156199397365544615530035232, −5.66249617915409273086294898323, −5.44821123705890341560842717593, −5.09378139017334053161677800056, −4.80639147457528683836584717368, −4.15595609768175443976298305622, −3.73197264380886513699184592884, −3.50259086024534713934440278145, −2.87233602967682451436281993917, −1.91239697986041212921589527643, −1.85912610215612717847088602238, −1.56376177442796804589164893808, −0.32807845338339505349024143005, 0.32807845338339505349024143005, 1.56376177442796804589164893808, 1.85912610215612717847088602238, 1.91239697986041212921589527643, 2.87233602967682451436281993917, 3.50259086024534713934440278145, 3.73197264380886513699184592884, 4.15595609768175443976298305622, 4.80639147457528683836584717368, 5.09378139017334053161677800056, 5.44821123705890341560842717593, 5.66249617915409273086294898323, 6.18156199397365544615530035232, 7.07659731465896497768463635599, 7.08284676861751351858076882137, 7.19035527388077362313746820214, 7.73036194167570246420712176758, 8.330648532192857778115414626618, 8.647074114512688486035137679657, 8.811972540626731772952779411353

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