Properties

Degree 4
Conductor $ 2^{12} \cdot 5^{2} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s − 50·5-s − 104·7-s − 158·9-s + 320·11-s + 100·13-s − 400·15-s + 580·17-s + 720·19-s − 832·21-s − 1.68e3·23-s + 1.87e3·25-s − 1.09e3·27-s − 108·29-s − 9.84e3·31-s + 2.56e3·33-s + 5.20e3·35-s − 6.54e3·37-s + 800·39-s − 1.06e4·41-s + 2.56e4·43-s + 7.90e3·45-s − 2.82e4·47-s − 2.52e4·49-s + 4.64e3·51-s − 3.13e4·53-s − 1.60e4·55-s + ⋯
L(s)  = 1  + 0.513·3-s − 0.894·5-s − 0.802·7-s − 0.650·9-s + 0.797·11-s + 0.164·13-s − 0.459·15-s + 0.486·17-s + 0.457·19-s − 0.411·21-s − 0.665·23-s + 3/5·25-s − 0.289·27-s − 0.0238·29-s − 1.83·31-s + 0.409·33-s + 0.717·35-s − 0.785·37-s + 0.0842·39-s − 0.986·41-s + 2.11·43-s + 0.581·45-s − 1.86·47-s − 1.50·49-s + 0.249·51-s − 1.53·53-s − 0.713·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(102400\)    =    \(2^{12} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{320} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 102400,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_1$ \( ( 1 + p^{2} T )^{2} \)
good3$D_{4}$ \( 1 - 8 T + 74 p T^{2} - 8 p^{5} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 + 104 T + 36038 T^{2} + 104 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 320 T + 319702 T^{2} - 320 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 100 T + 297086 T^{2} - 100 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 580 T + 2475814 T^{2} - 580 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 720 T + 4633798 T^{2} - 720 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 1688 T + 10950502 T^{2} + 1688 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 108 T + 39233214 T^{2} + 108 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 9840 T + 76732702 T^{2} + 9840 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 6540 T + 61572814 T^{2} + 6540 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 10620 T + 77106582 T^{2} + 10620 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 25672 T + 364912302 T^{2} - 25672 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 28296 T + 617782998 T^{2} + 28296 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 31340 T + 755347886 T^{2} + 31340 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 30800 T + 1666896598 T^{2} - 30800 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 24540 T + 1447225822 T^{2} + 24540 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 34584 T + 2930656478 T^{2} - 34584 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 12400 T + 1143670702 T^{2} - 12400 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 7180 T + 284279286 T^{2} + 7180 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 71840 T + 7430807198 T^{2} + 71840 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 31928 T + 5910993662 T^{2} + 31928 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 40748 T + 8570866774 T^{2} + 40748 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 190140 T + 19653817414 T^{2} + 190140 p^{5} T^{3} + p^{10} T^{4} \)
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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

−10.62987837960466234569276205775, −9.971023163662618109720797370292, −9.445631264284278534338581884381, −9.313796199708606407184527736847, −8.574194920765514380718969374936, −8.259546877681041866157767966972, −7.77707498899190618706917304721, −7.24348414993476455209093194515, −6.67408753355250997570310664551, −6.31467427659840304162269792085, −5.52541761282739948731512021685, −5.17486646166441525351350876396, −4.17105650498536724799991153446, −3.85588228745029694147188347257, −3.12846535711795993808237160403, −3.00888426324245440844702291400, −1.85988832960724002772026442577, −1.23616585723922726993668273931, 0, 0, 1.23616585723922726993668273931, 1.85988832960724002772026442577, 3.00888426324245440844702291400, 3.12846535711795993808237160403, 3.85588228745029694147188347257, 4.17105650498536724799991153446, 5.17486646166441525351350876396, 5.52541761282739948731512021685, 6.31467427659840304162269792085, 6.67408753355250997570310664551, 7.24348414993476455209093194515, 7.77707498899190618706917304721, 8.259546877681041866157767966972, 8.574194920765514380718969374936, 9.313796199708606407184527736847, 9.445631264284278534338581884381, 9.971023163662618109720797370292, 10.62987837960466234569276205775

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