Properties

Label 4-504e2-1.1-c3e2-0-9
Degree $4$
Conductor $254016$
Sign $1$
Analytic cond. $884.286$
Root an. cond. $5.45316$
Motivic weight $3$
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  − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(254016\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(884.286\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{504} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 254016,\ (\ :3/2, 3/2),\ 1)\)

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

−10.25766199974540252842845590230, −10.09046461708136248517122784480, −9.212162463562009456857222326841, −8.872034337651921528095099355461, −8.267781750084259463578303960035, −8.227884692787713696896016997207, −7.50628729151117848897611452819, −7.37854139940838831305789315317, −6.50140004726001976394573395993, −6.29890352925165513224438258557, −5.29866504147081077703908520838, −5.21944881798502008165559737448, −4.43308753737450677353002831174, −3.83930087190104098589088187740, −3.62111958497446769258211289943, −2.85074348449569151610321471841, −1.83185373793504048724206216768, −1.48465324693089921411677507212, 0, 0, 1.48465324693089921411677507212, 1.83185373793504048724206216768, 2.85074348449569151610321471841, 3.62111958497446769258211289943, 3.83930087190104098589088187740, 4.43308753737450677353002831174, 5.21944881798502008165559737448, 5.29866504147081077703908520838, 6.29890352925165513224438258557, 6.50140004726001976394573395993, 7.37854139940838831305789315317, 7.50628729151117848897611452819, 8.227884692787713696896016997207, 8.267781750084259463578303960035, 8.872034337651921528095099355461, 9.212162463562009456857222326841, 10.09046461708136248517122784480, 10.25766199974540252842845590230

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