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 − 0.0952·21-s + 9/23·23-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  =  \((4,\ 50176,\ (\ :1, 1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(3.95447\)
\(L(\frac12)\)  \(\approx\)  \(3.95447\)
\(L(2)\)   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

−12.29966507003708502875774346575, −11.98183841137149583367665519935, −11.22724074280459186907375695752, −10.72262484794728323379598385326, −10.01878498109662382281425206648, −9.754337153242918056491548524204, −9.565687483787918082469969074042, −9.109365318132976808602385263750, −8.229107741081319227184579666243, −7.998179545109023527211861883267, −6.87289950322865836149543944103, −6.77296967207349430006778859810, −6.06632152671562917702493855584, −5.79338405869396939118488398868, −4.75519084767241954599407174703, −4.43468771267298504794157420042, −3.34837268940314721058804358606, −2.85692819813363375503043366659, −1.49247896338481340627349212567, −1.47837986228554648595262899203, 1.47837986228554648595262899203, 1.49247896338481340627349212567, 2.85692819813363375503043366659, 3.34837268940314721058804358606, 4.43468771267298504794157420042, 4.75519084767241954599407174703, 5.79338405869396939118488398868, 6.06632152671562917702493855584, 6.77296967207349430006778859810, 6.87289950322865836149543944103, 7.998179545109023527211861883267, 8.229107741081319227184579666243, 9.109365318132976808602385263750, 9.565687483787918082469969074042, 9.754337153242918056491548524204, 10.01878498109662382281425206648, 10.72262484794728323379598385326, 11.22724074280459186907375695752, 11.98183841137149583367665519935, 12.29966507003708502875774346575

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