Properties

Degree 4
Conductor $ 2^{10} \cdot 7^{2} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·7-s − 10·9-s + 20·23-s + 22·25-s + 147·49-s − 140·63-s + 220·71-s − 260·79-s + 19·81-s − 52·113-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 280·161-s + 163-s + 167-s + 310·169-s + 173-s + 308·175-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2·7-s − 1.11·9-s + 0.869·23-s + 0.879·25-s + 3·49-s − 2.22·63-s + 3.09·71-s − 3.29·79-s + 0.234·81-s − 0.460·113-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 1.73·161-s + 0.00613·163-s + 0.00598·167-s + 1.83·169-s + 0.00578·173-s + 1.75·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(50176\)    =    \(2^{10} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 50176,\ (\ :1, 1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.38121\)
\(L(\frac12)\)  \(\approx\)  \(2.38121\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7$C_1$ \( ( 1 - p T )^{2} \)
good3$C_2^2$ \( 1 + 10 T^{2} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 22 T^{2} + p^{4} T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
13$C_2^2$ \( 1 - 310 T^{2} + p^{4} T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
19$C_2^2$ \( 1 + 650 T^{2} + p^{4} T^{4} \)
23$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
59$C_2^2$ \( 1 + 1130 T^{2} + p^{4} T^{4} \)
61$C_2^2$ \( 1 + 7370 T^{2} + p^{4} T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
71$C_2$ \( ( 1 - 110 T + p^{2} T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
79$C_2$ \( ( 1 + 130 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 13130 T^{2} + p^{4} T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
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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

−12.17908525126606484546904293654, −11.57747829660469868553873422623, −11.36595371442362218026199338239, −10.86937113443377711471404444849, −10.64492958306335276113964921983, −9.835743042178350734614050249043, −9.204737382735169592152918810480, −8.703658557950374757305517657244, −8.192562701493007242981464210286, −8.130822355328534881737698920012, −7.08435692803129348903373701827, −6.99898681361773318197710320373, −5.72922778543844640691800002524, −5.67167583127100728382666263997, −4.74719880245968070293012242350, −4.58518661856689977379358086926, −3.51027084711977552061963525155, −2.70912996390944121597786294158, −1.91221612892844471889225483613, −0.902089358644509428602348063629, 0.902089358644509428602348063629, 1.91221612892844471889225483613, 2.70912996390944121597786294158, 3.51027084711977552061963525155, 4.58518661856689977379358086926, 4.74719880245968070293012242350, 5.67167583127100728382666263997, 5.72922778543844640691800002524, 6.99898681361773318197710320373, 7.08435692803129348903373701827, 8.130822355328534881737698920012, 8.192562701493007242981464210286, 8.703658557950374757305517657244, 9.204737382735169592152918810480, 9.835743042178350734614050249043, 10.64492958306335276113964921983, 10.86937113443377711471404444849, 11.36595371442362218026199338239, 11.57747829660469868553873422623, 12.17908525126606484546904293654

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