Properties

Label 4-539e2-1.1-c3e2-0-0
Degree $4$
Conductor $290521$
Sign $1$
Analytic cond. $1011.36$
Root an. cond. $5.63932$
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  + 2·2-s + 2·3-s − 10·4-s − 2·5-s + 4·6-s − 32·8-s − 3·9-s − 4·10-s − 22·11-s − 20·12-s − 80·13-s − 4·15-s + 44·16-s + 124·17-s − 6·18-s − 72·19-s + 20·20-s − 44·22-s − 98·23-s − 64·24-s − 55·25-s − 160·26-s + 34·27-s + 144·29-s − 8·30-s + 34·31-s + 248·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.384·3-s − 5/4·4-s − 0.178·5-s + 0.272·6-s − 1.41·8-s − 1/9·9-s − 0.126·10-s − 0.603·11-s − 0.481·12-s − 1.70·13-s − 0.0688·15-s + 0.687·16-s + 1.76·17-s − 0.0785·18-s − 0.869·19-s + 0.223·20-s − 0.426·22-s − 0.888·23-s − 0.544·24-s − 0.439·25-s − 1.20·26-s + 0.242·27-s + 0.922·29-s − 0.0486·30-s + 0.196·31-s + 1.37·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(290521\)    =    \(7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1011.36\)
Root analytic conductor: \(5.63932\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{539} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 290521,\ (\ :3/2, 3/2),\ 1)\)

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)$
bad7 \( 1 \)
11$C_1$ \( ( 1 + p T )^{2} \)
good2$D_{4}$ \( 1 - p T + 7 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
3$D_{4}$ \( 1 - 2 T + 7 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 59 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 80 T + 4794 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 124 T + 13238 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 72 T + 4214 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 98 T + 22847 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 144 T + 44554 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 34 T + 57519 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 54 T + 101843 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 536 T + 209618 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 60 T + 159146 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 272 T + 182942 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 492 T + 348862 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 634 T + 458975 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 840 T + 528794 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 754 T + 742455 T^{2} - 754 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 678 T + 813415 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 400 T + 160962 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 4 p T - 279966 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 468 T + 1155130 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1842 T + 1935427 T^{2} - 1842 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 2194 T + 2966547 T^{2} + 2194 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.971991675506078997338983093270, −9.807914654316438045463829338871, −9.454726767681092905591339790584, −8.809421479825941995080082368838, −8.226300289574639172701729561315, −8.204192816959429673320366388741, −7.52993170768753087772705693121, −7.20122426204243107397606490682, −6.24886501379630788260314473550, −5.99425873795175285840569123791, −5.27460625594674344609724228187, −4.80093132741622055213586054978, −4.72968351477440373709047311640, −3.92858771523133254829491865070, −3.41282771992919602999633646117, −2.91691763428118188016738661083, −2.26688064799501805895340432730, −1.25927469338755089467574721630, 0, 0, 1.25927469338755089467574721630, 2.26688064799501805895340432730, 2.91691763428118188016738661083, 3.41282771992919602999633646117, 3.92858771523133254829491865070, 4.72968351477440373709047311640, 4.80093132741622055213586054978, 5.27460625594674344609724228187, 5.99425873795175285840569123791, 6.24886501379630788260314473550, 7.20122426204243107397606490682, 7.52993170768753087772705693121, 8.204192816959429673320366388741, 8.226300289574639172701729561315, 8.809421479825941995080082368838, 9.454726767681092905591339790584, 9.807914654316438045463829338871, 9.971991675506078997338983093270

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