Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{4} \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  − 6·5-s + 7-s + 6·11-s + 5·19-s − 12·23-s + 19·25-s − 5·31-s − 6·35-s + 11·37-s + 6·47-s − 6·49-s + 12·53-s − 36·55-s − 12·59-s − 24·61-s + 15·67-s − 9·73-s + 6·77-s + 21·79-s + 36·83-s − 12·89-s − 30·95-s − 18·101-s − 5·103-s + 7·109-s + 12·113-s + 72·115-s + ⋯
L(s)  = 1  − 2.68·5-s + 0.377·7-s + 1.80·11-s + 1.14·19-s − 2.50·23-s + 19/5·25-s − 0.898·31-s − 1.01·35-s + 1.80·37-s + 0.875·47-s − 6/7·49-s + 1.64·53-s − 4.85·55-s − 1.56·59-s − 3.07·61-s + 1.83·67-s − 1.05·73-s + 0.683·77-s + 2.36·79-s + 3.95·83-s − 1.27·89-s − 3.07·95-s − 1.79·101-s − 0.492·103-s + 0.670·109-s + 1.12·113-s + 6.71·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.024285097\)
\(L(\frac12)\)  \(\approx\)  \(1.024285097\)
\(L(\frac{3}{2})\)   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$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \)
41$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−10.24202483320007231712126742819, −9.513056003111193065572896681447, −9.328350962423188572928703700507, −8.997927582986173972982356615092, −8.179619022526110775691749660739, −8.097822301255009424413827706456, −7.73280869412373413483983015680, −7.43319898673919815538413594361, −6.97363138599567634218341814174, −6.32234341339727357080220407994, −6.09252956179449645909593151779, −5.36500534983608535166264534980, −4.61743079853058161984363137300, −4.34882733225343345910354384387, −3.87185642525035786147726792906, −3.66220981038429346387326354516, −3.17003521237528029480590233449, −2.19679293088573054801475842264, −1.32996170163245148892856759690, −0.51546316417718536849644538196, 0.51546316417718536849644538196, 1.32996170163245148892856759690, 2.19679293088573054801475842264, 3.17003521237528029480590233449, 3.66220981038429346387326354516, 3.87185642525035786147726792906, 4.34882733225343345910354384387, 4.61743079853058161984363137300, 5.36500534983608535166264534980, 6.09252956179449645909593151779, 6.32234341339727357080220407994, 6.97363138599567634218341814174, 7.43319898673919815538413594361, 7.73280869412373413483983015680, 8.097822301255009424413827706456, 8.179619022526110775691749660739, 8.997927582986173972982356615092, 9.328350962423188572928703700507, 9.513056003111193065572896681447, 10.24202483320007231712126742819

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