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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 9·5-s + 2·7-s + 9·9-s + 17·11-s − 9·15-s + 25·17-s − 7·19-s + 2·21-s − 9·23-s + 29·25-s + 26·27-s − 57·31-s + 17·33-s − 18·35-s + 15·37-s + 52·41-s − 28·43-s − 81·45-s + 87·47-s − 45·49-s + 25·51-s + 159·53-s − 153·55-s − 7·57-s − 55·59-s + 39·61-s + ⋯
L(s)  = 1  + 1/3·3-s − 9/5·5-s + 2/7·7-s + 9-s + 1.54·11-s − 3/5·15-s + 1.47·17-s − 0.368·19-s + 2/21·21-s − 0.391·23-s + 1.15·25-s + 0.962·27-s − 1.83·31-s + 0.515·33-s − 0.514·35-s + 0.405·37-s + 1.26·41-s − 0.651·43-s − 9/5·45-s + 1.85·47-s − 0.918·49-s + 0.490·51-s + 3·53-s − 2.78·55-s − 0.122·57-s − 0.932·59-s + 0.639·61-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  =  \((4,\ 50176,\ (\ :1, 1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.93156\)
\(L(\frac12)\)  \(\approx\)  \(1.93156\)
\(L(2)\)   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$C_2$ \( 1 - 2 T + p^{2} T^{2} \)
good3$C_2^2$ \( 1 - T - 8 T^{2} - p^{2} T^{3} + p^{4} T^{4} \)
5$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 17 T + 168 T^{2} - 17 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \)
17$C_2^2$ \( 1 - 25 T + 336 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2^2$ \( 1 + 7 T - 312 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} \)
23$C_2^2$ \( 1 + 9 T + 556 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1490 T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 + 57 T + 2044 T^{2} + 57 p^{2} T^{3} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 15 T + 1444 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 87 T + 4732 T^{2} - 87 p^{2} T^{3} + p^{4} T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \)
59$C_2^2$ \( 1 + 55 T - 456 T^{2} + 55 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 39 T + 4228 T^{2} - 39 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 17 T - 4200 T^{2} - 17 p^{2} T^{3} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 + 119 T + 8832 T^{2} + 119 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2^2$ \( 1 + 129 T + 11788 T^{2} + 129 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2$ \( ( 1 + 110 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 71 T - 2880 T^{2} + 71 p^{2} T^{3} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{2} \)
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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

−12.25483576392990366429238027063, −11.60301348168794433735797890355, −11.59329595281554338129202678429, −10.99379657716838765626466411014, −10.11402219783600085443445947050, −10.07795307176258311849264804878, −9.212905316801777942365979451242, −8.662716345705989752133331530599, −8.449809937801265543653716149277, −7.56737398640980444868945323185, −7.23933598184862773709201889110, −7.19416498957755656186953152040, −6.07851328555039675789654090660, −5.58996298332999203254156216581, −4.50761519667970612628382964577, −4.09144513729165110344838127044, −3.81447729561039505965949254394, −3.05667365462859865358890721342, −1.74670920262130021193647966917, −0.815849817403449677751324828106, 0.815849817403449677751324828106, 1.74670920262130021193647966917, 3.05667365462859865358890721342, 3.81447729561039505965949254394, 4.09144513729165110344838127044, 4.50761519667970612628382964577, 5.58996298332999203254156216581, 6.07851328555039675789654090660, 7.19416498957755656186953152040, 7.23933598184862773709201889110, 7.56737398640980444868945323185, 8.449809937801265543653716149277, 8.662716345705989752133331530599, 9.212905316801777942365979451242, 10.07795307176258311849264804878, 10.11402219783600085443445947050, 10.99379657716838765626466411014, 11.59329595281554338129202678429, 11.60301348168794433735797890355, 12.25483576392990366429238027063

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