Properties

Label 4-1008e2-1.1-c5e2-0-11
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $26136.1$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 78·5-s − 98·7-s + 174·11-s + 208·13-s − 1.48e3·17-s − 352·19-s + 3.35e3·23-s + 2.85e3·25-s − 276·29-s − 6.52e3·31-s + 7.64e3·35-s + 1.38e4·37-s + 1.29e4·41-s − 1.27e4·43-s − 2.81e4·47-s + 7.20e3·49-s + 4.69e4·53-s − 1.35e4·55-s − 6.55e4·59-s − 1.31e4·61-s − 1.62e4·65-s + 7.52e4·67-s − 6.60e4·71-s + 6.04e4·73-s − 1.70e4·77-s + 3.49e4·79-s − 8.24e4·83-s + ⋯
L(s)  = 1  − 1.39·5-s − 0.755·7-s + 0.433·11-s + 0.341·13-s − 1.24·17-s − 0.223·19-s + 1.32·23-s + 0.914·25-s − 0.0609·29-s − 1.21·31-s + 1.05·35-s + 1.66·37-s + 1.20·41-s − 1.04·43-s − 1.85·47-s + 3/7·49-s + 2.29·53-s − 0.604·55-s − 2.45·59-s − 0.452·61-s − 0.476·65-s + 2.04·67-s − 1.55·71-s + 1.32·73-s − 0.327·77-s + 0.629·79-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26136.1\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1016064,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 + p^{2} T )^{2} \)
good5$D_{4}$ \( 1 + 78 T + 3226 T^{2} + 78 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 174 T + 216046 T^{2} - 174 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 16 p T + 589782 T^{2} - 16 p^{6} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 1482 T + 1748050 T^{2} + 1482 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 352 T - 907146 T^{2} + 352 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 3354 T + 13680670 T^{2} - 3354 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 276 T - 4408658 T^{2} + 276 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 6520 T + 61995582 T^{2} + 6520 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 13864 T + 182650038 T^{2} - 13864 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 12930 T + 272958682 T^{2} - 12930 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 12712 T + 268967622 T^{2} + 12712 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 28116 T + 641027998 T^{2} + 28116 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 46992 T + 1329386182 T^{2} - 46992 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 65556 T + 2467431382 T^{2} + 65556 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 13148 T + 1323360078 T^{2} + 13148 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 75236 T + 4068077958 T^{2} - 75236 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 66042 T + 4539911038 T^{2} + 66042 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 60496 T + 4988928270 T^{2} - 60496 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 34916 T + 3983487582 T^{2} - 34916 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 82488 T + 9550060822 T^{2} + 82488 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 42510 T + 4392793978 T^{2} + 42510 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 213256 T + 28472054478 T^{2} - 213256 p^{5} T^{3} + p^{10} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.019304493183041706347359607186, −8.673426650838941802186296450483, −8.051104124478749091926629119294, −7.933936027181014797261657516363, −7.22114167522106885735100642756, −6.99958280704156735019284454360, −6.55974272170554742626207304015, −6.17849038766310963069514310135, −5.62445925909879558583495406493, −5.00422136167680257886338900566, −4.49046138470818933952966955484, −4.20027496028757755077374614581, −3.58887889145828041099398340385, −3.37137892573862412966444770528, −2.68470075078786629601185064226, −2.21441376989172694708514872592, −1.33628438909052813276167543332, −0.861559162196326740773649012234, 0, 0, 0.861559162196326740773649012234, 1.33628438909052813276167543332, 2.21441376989172694708514872592, 2.68470075078786629601185064226, 3.37137892573862412966444770528, 3.58887889145828041099398340385, 4.20027496028757755077374614581, 4.49046138470818933952966955484, 5.00422136167680257886338900566, 5.62445925909879558583495406493, 6.17849038766310963069514310135, 6.55974272170554742626207304015, 6.99958280704156735019284454360, 7.22114167522106885735100642756, 7.933936027181014797261657516363, 8.051104124478749091926629119294, 8.673426650838941802186296450483, 9.019304493183041706347359607186

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