Properties

Label 4-1008e2-1.1-c3e2-0-12
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $3537.14$
Root an. cond. $7.71193$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·5-s − 14·7-s + 18·11-s + 48·13-s − 34·17-s + 16·19-s + 110·23-s + 74·25-s − 212·29-s + 136·31-s + 196·35-s − 24·37-s − 694·41-s + 584·43-s − 316·47-s + 147·49-s − 560·53-s − 252·55-s − 492·59-s − 604·61-s − 672·65-s + 1.02e3·67-s − 1.71e3·71-s − 1.31e3·73-s − 252·77-s + 556·79-s − 264·83-s + ⋯
L(s)  = 1  − 1.25·5-s − 0.755·7-s + 0.493·11-s + 1.02·13-s − 0.485·17-s + 0.193·19-s + 0.997·23-s + 0.591·25-s − 1.35·29-s + 0.787·31-s + 0.946·35-s − 0.106·37-s − 2.64·41-s + 2.07·43-s − 0.980·47-s + 3/7·49-s − 1.45·53-s − 0.617·55-s − 1.08·59-s − 1.26·61-s − 1.28·65-s + 1.85·67-s − 2.85·71-s − 2.10·73-s − 0.372·77-s + 0.791·79-s − 0.349·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3537.14\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1016064,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
good5$D_{4}$ \( 1 + 14 T + 122 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 18 T + 1150 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 48 T + 4262 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 2 p T - 4222 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 16 T + 10950 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 110 T + 18686 T^{2} - 110 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 212 T + 57182 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 136 T + 61374 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 24 T - 18202 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 694 T + 243914 T^{2} + 694 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 584 T + 232950 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 316 T + 231902 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 560 T + 369782 T^{2} + 560 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 492 T + 351622 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 604 T + 406398 T^{2} + 604 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 1020 T + 843926 T^{2} - 1020 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1710 T + 1336222 T^{2} + 1710 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1312 T + 1201998 T^{2} + 1312 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 556 T + 751134 T^{2} - 556 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 264 T + 979750 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 70 T - 186262 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 136 T + 1812270 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.236378989940766534048141790688, −8.967278705808705292933685160202, −8.557497873834533495646956044863, −8.222978874028214681779156202897, −7.56470413679584112542252109108, −7.36882942875971443385206816302, −6.89125290866406970304454002137, −6.39188239121309759532992275516, −6.06159912957103008621050021437, −5.59078286862576900355894163018, −4.72911194119947283309987660761, −4.60665405853406483759528302819, −3.90997272487971659521127227672, −3.42008531388878664600426485287, −3.26199141602226603449910781396, −2.53980732672272079458197846216, −1.55937563887268020883397695451, −1.15314603450704085564718602622, 0, 0, 1.15314603450704085564718602622, 1.55937563887268020883397695451, 2.53980732672272079458197846216, 3.26199141602226603449910781396, 3.42008531388878664600426485287, 3.90997272487971659521127227672, 4.60665405853406483759528302819, 4.72911194119947283309987660761, 5.59078286862576900355894163018, 6.06159912957103008621050021437, 6.39188239121309759532992275516, 6.89125290866406970304454002137, 7.36882942875971443385206816302, 7.56470413679584112542252109108, 8.222978874028214681779156202897, 8.557497873834533495646956044863, 8.967278705808705292933685160202, 9.236378989940766534048141790688

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